<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Linear algebra — Julia Language 0.3.4 documentation</title>
<link href='https://fonts.googleapis.com/css?family=Lato:400,700|Roboto+Slab:400,700|Inconsolata:400,700' rel='stylesheet' type='text/css'>
<link rel="stylesheet" href="../_static/julia.css" type="text/css" />
<link rel="stylesheet" href="../_static/julia.css" type="text/css" />
<link rel="top" title="Julia Language 0.3.4 documentation" href="../index.html"/>
<link rel="next" title="Networking and Streams" href="networking-and-streams.html"/>
<link rel="prev" title="Multi-dimensional Arrays" href="arrays.html"/>
<script src="https://cdnjs.cloudflare.com/ajax/libs/modernizr/2.6.2/modernizr.min.js"></script>
</head>
<body class="wy-body-for-nav" role="document">
<div class="wy-grid-for-nav">
<nav data-toggle="wy-nav-shift" class="wy-nav-side">
<div class="wy-side-nav-search">
<a href="http://julialang.org/"><img src="../_static/julia-logo.svg" class="logo"></a>
<!--
<a href="../index.html" class="fa fa-home"> Julia Language</a>
-->
<div role="search">
<form id ="rtd-search-form" class="wy-form" action="../search.html" method="get">
<input type="text" name="q" placeholder="Search docs" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
<div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="introduction.html">Introduction</a></li>
<li class="toctree-l1"><a class="reference internal" href="getting-started.html">Getting Started</a><ul>
<li class="toctree-l2"><a class="reference internal" href="getting-started.html#resources">Resources</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="variables.html">Variables</a><ul>
<li class="toctree-l2"><a class="reference internal" href="variables.html#allowed-variable-names">Allowed Variable Names</a></li>
<li class="toctree-l2"><a class="reference internal" href="variables.html#stylistic-conventions">Stylistic Conventions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="integers-and-floating-point-numbers.html">Integers and Floating-Point Numbers</a><ul>
<li class="toctree-l2"><a class="reference internal" href="integers-and-floating-point-numbers.html#integers">Integers</a></li>
<li class="toctree-l2"><a class="reference internal" href="integers-and-floating-point-numbers.html#floating-point-numbers">Floating-Point Numbers</a></li>
<li class="toctree-l2"><a class="reference internal" href="integers-and-floating-point-numbers.html#arbitrary-precision-arithmetic">Arbitrary Precision Arithmetic</a></li>
<li class="toctree-l2"><a class="reference internal" href="integers-and-floating-point-numbers.html#numeric-literal-coefficients">Numeric Literal Coefficients</a></li>
<li class="toctree-l2"><a class="reference internal" href="integers-and-floating-point-numbers.html#literal-zero-and-one">Literal zero and one</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="mathematical-operations.html">Mathematical Operations and Elementary Functions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="mathematical-operations.html#arithmetic-operators">Arithmetic Operators</a></li>
<li class="toctree-l2"><a class="reference internal" href="mathematical-operations.html#bitwise-operators">Bitwise Operators</a></li>
<li class="toctree-l2"><a class="reference internal" href="mathematical-operations.html#updating-operators">Updating operators</a></li>
<li class="toctree-l2"><a class="reference internal" href="mathematical-operations.html#numeric-comparisons">Numeric Comparisons</a></li>
<li class="toctree-l2"><a class="reference internal" href="mathematical-operations.html#elementary-functions">Elementary Functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="complex-and-rational-numbers.html">Complex and Rational Numbers</a><ul>
<li class="toctree-l2"><a class="reference internal" href="complex-and-rational-numbers.html#complex-numbers">Complex Numbers</a></li>
<li class="toctree-l2"><a class="reference internal" href="complex-and-rational-numbers.html#rational-numbers">Rational Numbers</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="strings.html">Strings</a><ul>
<li class="toctree-l2"><a class="reference internal" href="strings.html#characters">Characters</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#string-basics">String Basics</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#unicode-and-utf-8">Unicode and UTF-8</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#interpolation">Interpolation</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#common-operations">Common Operations</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#non-standard-string-literals">Non-Standard String Literals</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#regular-expressions">Regular Expressions</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#id3">Byte Array Literals</a></li>
<li class="toctree-l2"><a class="reference internal" href="strings.html#version-number-literals">Version Number Literals</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="functions.html">Functions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="functions.html#argument-passing-behavior">Argument Passing Behavior</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#the-return-keyword">The <tt class="docutils literal"><span class="pre">return</span></tt> Keyword</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#id1">Operators Are Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#operators-with-special-names">Operators With Special Names</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#anonymous-functions">Anonymous Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#multiple-return-values">Multiple Return Values</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#varargs-functions">Varargs Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#optional-arguments">Optional Arguments</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#keyword-arguments">Keyword Arguments</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#evaluation-scope-of-default-values">Evaluation Scope of Default Values</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#block-syntax-for-function-arguments">Block Syntax for Function Arguments</a></li>
<li class="toctree-l2"><a class="reference internal" href="functions.html#further-reading">Further Reading</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="control-flow.html">Control Flow</a><ul>
<li class="toctree-l2"><a class="reference internal" href="control-flow.html#compound-expressions">Compound Expressions</a></li>
<li class="toctree-l2"><a class="reference internal" href="control-flow.html#conditional-evaluation">Conditional Evaluation</a></li>
<li class="toctree-l2"><a class="reference internal" href="control-flow.html#short-circuit-evaluation">Short-Circuit Evaluation</a></li>
<li class="toctree-l2"><a class="reference internal" href="control-flow.html#repeated-evaluation-loops">Repeated Evaluation: Loops</a></li>
<li class="toctree-l2"><a class="reference internal" href="control-flow.html#exception-handling">Exception Handling</a></li>
<li class="toctree-l2"><a class="reference internal" href="control-flow.html#tasks-aka-coroutines">Tasks (aka Coroutines)</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="variables-and-scoping.html">Scope of Variables</a><ul>
<li class="toctree-l2"><a class="reference internal" href="variables-and-scoping.html#for-loops-and-comprehensions">For Loops and Comprehensions</a></li>
<li class="toctree-l2"><a class="reference internal" href="variables-and-scoping.html#constants">Constants</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="types.html">Types</a><ul>
<li class="toctree-l2"><a class="reference internal" href="types.html#type-declarations">Type Declarations</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#abstract-types">Abstract Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#bits-types">Bits Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#composite-types">Composite Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#immutable-composite-types">Immutable Composite Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#declared-types">Declared Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#tuple-types">Tuple Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#type-unions">Type Unions</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#man-parametric-types">Parametric Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#type-aliases">Type Aliases</a></li>
<li class="toctree-l2"><a class="reference internal" href="types.html#operations-on-types">Operations on Types</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="methods.html">Methods</a><ul>
<li class="toctree-l2"><a class="reference internal" href="methods.html#defining-methods">Defining Methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="methods.html#method-ambiguities">Method Ambiguities</a></li>
<li class="toctree-l2"><a class="reference internal" href="methods.html#parametric-methods">Parametric Methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="methods.html#note-on-optional-and-keyword-arguments">Note on Optional and keyword Arguments</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="constructors.html">Constructors</a><ul>
<li class="toctree-l2"><a class="reference internal" href="constructors.html#outer-constructor-methods">Outer Constructor Methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="constructors.html#inner-constructor-methods">Inner Constructor Methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="constructors.html#incomplete-initialization">Incomplete Initialization</a></li>
<li class="toctree-l2"><a class="reference internal" href="constructors.html#parametric-constructors">Parametric Constructors</a></li>
<li class="toctree-l2"><a class="reference internal" href="constructors.html#case-study-rational">Case Study: Rational</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="conversion-and-promotion.html">Conversion and Promotion</a><ul>
<li class="toctree-l2"><a class="reference internal" href="conversion-and-promotion.html#conversion">Conversion</a></li>
<li class="toctree-l2"><a class="reference internal" href="conversion-and-promotion.html#promotion">Promotion</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="modules.html">Modules</a><ul>
<li class="toctree-l2"><a class="reference internal" href="modules.html#summary-of-module-usage">Summary of module usage</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="metaprogramming.html">Metaprogramming</a><ul>
<li class="toctree-l2"><a class="reference internal" href="metaprogramming.html#expressions-and-eval">Expressions and Eval</a></li>
<li class="toctree-l2"><a class="reference internal" href="metaprogramming.html#macros">Macros</a></li>
<li class="toctree-l2"><a class="reference internal" href="metaprogramming.html#reflection">Reflection</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="arrays.html">Multi-dimensional Arrays</a><ul>
<li class="toctree-l2"><a class="reference internal" href="arrays.html#arrays">Arrays</a></li>
<li class="toctree-l2"><a class="reference internal" href="arrays.html#sparse-matrices">Sparse Matrices</a></li>
</ul>
</li>
<li class="toctree-l1 current"><a class="current reference internal" href="">Linear algebra</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#matrix-factorizations">Matrix factorizations</a></li>
<li class="toctree-l2"><a class="reference internal" href="#special-matrices">Special matrices</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="networking-and-streams.html">Networking and Streams</a><ul>
<li class="toctree-l2"><a class="reference internal" href="networking-and-streams.html#basic-stream-i-o">Basic Stream I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="networking-and-streams.html#text-i-o">Text I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="networking-and-streams.html#working-with-files">Working with Files</a></li>
<li class="toctree-l2"><a class="reference internal" href="networking-and-streams.html#a-simple-tcp-example">A simple TCP example</a></li>
<li class="toctree-l2"><a class="reference internal" href="networking-and-streams.html#resolving-ip-addresses">Resolving IP Addresses</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="parallel-computing.html">Parallel Computing</a><ul>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#data-movement">Data Movement</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#parallel-map-and-loops">Parallel Map and Loops</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#synchronization-with-remote-references">Synchronization With Remote References</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#scheduling">Scheduling</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#distributed-arrays">Distributed Arrays</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#constructing-distributed-arrays">Constructing Distributed Arrays</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#distributed-array-operations">Distributed Array Operations</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#shared-arrays-experimental">Shared Arrays (Experimental)</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel-computing.html#clustermanagers">ClusterManagers</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="running-external-programs.html">Running External Programs</a><ul>
<li class="toctree-l2"><a class="reference internal" href="running-external-programs.html#interpolation">Interpolation</a></li>
<li class="toctree-l2"><a class="reference internal" href="running-external-programs.html#quoting">Quoting</a></li>
<li class="toctree-l2"><a class="reference internal" href="running-external-programs.html#pipelines">Pipelines</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="calling-c-and-fortran-code.html">Calling C and Fortran Code</a><ul>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#mapping-c-types-to-julia">Mapping C Types to Julia</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#accessing-data-through-a-pointer">Accessing Data through a Pointer</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#passing-pointers-for-modifying-inputs">Passing Pointers for Modifying Inputs</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#garbage-collection-safety">Garbage Collection Safety</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#non-constant-function-specifications">Non-constant Function Specifications</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#indirect-calls">Indirect Calls</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#calling-convention">Calling Convention</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#accessing-global-variables">Accessing Global Variables</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#passing-julia-callback-functions-to-c">Passing Julia Callback Functions to C</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#c">C++</a></li>
<li class="toctree-l2"><a class="reference internal" href="calling-c-and-fortran-code.html#handling-platform-variations">Handling Platform Variations</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="interacting-with-julia.html">Interacting With Julia</a><ul>
<li class="toctree-l2"><a class="reference internal" href="interacting-with-julia.html#the-different-prompt-modes">The different prompt modes</a></li>
<li class="toctree-l2"><a class="reference internal" href="interacting-with-julia.html#key-bindings">Key bindings</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="embedding.html">Embedding Julia</a><ul>
<li class="toctree-l2"><a class="reference internal" href="embedding.html#high-level-embedding">High-Level Embedding</a></li>
<li class="toctree-l2"><a class="reference internal" href="embedding.html#converting-types">Converting Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="embedding.html#calling-julia-functions">Calling Julia Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="embedding.html#memory-management">Memory Management</a></li>
<li class="toctree-l2"><a class="reference internal" href="embedding.html#working-with-arrays">Working with Arrays</a></li>
<li class="toctree-l2"><a class="reference internal" href="embedding.html#exceptions">Exceptions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="packages.html">Packages</a><ul>
<li class="toctree-l2"><a class="reference internal" href="packages.html#package-status">Package Status</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#adding-and-removing-packages">Adding and Removing Packages</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#installing-unregistered-packages">Installing Unregistered Packages</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#updating-packages">Updating Packages</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#checkout-pin-and-free">Checkout, Pin and Free</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="packages.html#package-development">Package Development</a><ul>
<li class="toctree-l2"><a class="reference internal" href="packages.html#initial-setup">Initial Setup</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#generating-a-new-package">Generating a New Package</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#making-your-package-available">Making Your Package Available</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#publishing-your-package">Publishing Your Package</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#tagging-package-versions">Tagging Package Versions</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#fixing-package-requirements">Fixing Package Requirements</a></li>
<li class="toctree-l2"><a class="reference internal" href="packages.html#man-package-requirements">Requirements Specification</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="performance-tips.html">Performance Tips</a><ul>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#avoid-global-variables">Avoid global variables</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#measure-performance-with-time-and-pay-attention-to-memory-allocation">Measure performance with <tt class="docutils literal"><span class="pre">@time</span></tt> and pay attention to memory allocation</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#tools">Tools</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#avoid-containers-with-abstract-type-parameters">Avoid containers with abstract type parameters</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#type-declarations">Type declarations</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#break-functions-into-multiple-definitions">Break functions into multiple definitions</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#write-type-stable-functions">Write “type-stable” functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#avoid-changing-the-type-of-a-variable">Avoid changing the type of a variable</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#separate-kernel-functions">Separate kernel functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#access-arrays-in-memory-order-along-columns">Access arrays in memory order, along columns</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#pre-allocating-outputs">Pre-allocating outputs</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#avoid-string-interpolation-for-i-o">Avoid string interpolation for I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#fix-deprecation-warnings">Fix deprecation warnings</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#tweaks">Tweaks</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance-tips.html#performance-annotations">Performance Annotations</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="style-guide.html">Style Guide</a><ul>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#write-functions-not-just-scripts">Write functions, not just scripts</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#avoid-writing-overly-specific-types">Avoid writing overly-specific types</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#handle-excess-argument-diversity-in-the-caller">Handle excess argument diversity in the caller</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#append-to-names-of-functions-that-modify-their-arguments">Append <cite>!</cite> to names of functions that modify their arguments</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#avoid-strange-type-unions">Avoid strange type Unions</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#try-to-avoid-nullable-fields">Try to avoid nullable fields</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#avoid-elaborate-container-types">Avoid elaborate container types</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#avoid-underscores-in-names">Avoid underscores in names</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-overuse-try-catch">Don’t overuse try-catch</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-parenthesize-conditions">Don’t parenthesize conditions</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-overuse">Don’t overuse ...</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-use-unnecessary-static-parameters">Don’t use unnecessary static parameters</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#avoid-confusion-about-whether-something-is-an-instance-or-a-type">Avoid confusion about whether something is an instance or a type</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-overuse-macros">Don’t overuse macros</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-expose-unsafe-operations-at-the-interface-level">Don’t expose unsafe operations at the interface level</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#don-t-overload-methods-of-base-container-types">Don’t overload methods of base container types</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#be-careful-with-type-equality">Be careful with type equality</a></li>
<li class="toctree-l2"><a class="reference internal" href="style-guide.html#do-not-write-x-f-x">Do not write <tt class="docutils literal"><span class="pre">x->f(x)</span></tt></a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="faq.html">Frequently Asked Questions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="faq.html#sessions-and-the-repl">Sessions and the REPL</a></li>
<li class="toctree-l2"><a class="reference internal" href="faq.html#functions">Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="faq.html#types-type-declarations-and-constructors">Types, type declarations, and constructors</a></li>
<li class="toctree-l2"><a class="reference internal" href="faq.html#nothingness-and-missing-values">Nothingness and missing values</a></li>
<li class="toctree-l2"><a class="reference internal" href="faq.html#julia-releases">Julia Releases</a></li>
<li class="toctree-l2"><a class="reference internal" href="faq.html#developing-julia">Developing Julia</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="noteworthy-differences.html">Noteworthy Differences from other Languages</a><ul>
<li class="toctree-l2"><a class="reference internal" href="noteworthy-differences.html#noteworthy-differences-from-matlab">Noteworthy differences from MATLAB</a></li>
<li class="toctree-l2"><a class="reference internal" href="noteworthy-differences.html#noteworthy-differences-from-r">Noteworthy differences from R</a></li>
<li class="toctree-l2"><a class="reference internal" href="noteworthy-differences.html#noteworthy-differences-from-python">Noteworthy differences from Python</a></li>
</ul>
</li>
</ul>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/base.html">The Standard Library</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#introduction">Introduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#getting-around">Getting Around</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#all-objects">All Objects</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#types">Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#generic-functions">Generic Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#syntax">Syntax</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#iteration">Iteration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#general-collections">General Collections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#iterable-collections">Iterable Collections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#indexable-collections">Indexable Collections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#associative-collections">Associative Collections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#set-like-collections">Set-Like Collections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#dequeues">Dequeues</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#strings">Strings</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#i-o">I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#network-i-o">Network I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#text-i-o">Text I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#multimedia-i-o">Multimedia I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#memory-mapped-i-o">Memory-mapped I/O</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#standard-numeric-types">Standard Numeric Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#mathematical-operators">Mathematical Operators</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#mathematical-functions">Mathematical Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#data-formats">Data Formats</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#numbers">Numbers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#bigfloats">BigFloats</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#random-numbers">Random Numbers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#arrays">Arrays</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#combinatorics">Combinatorics</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#statistics">Statistics</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#signal-processing">Signal Processing</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#numerical-integration">Numerical Integration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#parallel-computing">Parallel Computing</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#distributed-arrays">Distributed Arrays</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#shared-arrays-experimental-unix-only-feature">Shared Arrays (Experimental, UNIX-only feature)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#system">System</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#c-interface">C Interface</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#errors">Errors</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#tasks">Tasks</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#events">Events</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#reflection">Reflection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/base.html#internals">Internals</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/sparse.html">Sparse Matrices</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/linalg.html">Linear Algebra</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/linalg.html#module-Base.LinAlg.BLAS">BLAS Functions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/constants.html">Constants</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/file.html">Filesystem</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/punctuation.html">Punctuation</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/sort.html">Sorting and Related Functions</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/sort.html#sorting-functions">Sorting Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/sort.html#order-related-functions">Order-Related Functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/sort.html#sorting-algorithms">Sorting Algorithms</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/pkg.html">Package Manager Functions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/collections.html">Collections and Data Structures</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/collections.html#priorityqueue">PriorityQueue</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/collections.html#heap-functions">Heap Functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/graphics.html">Graphics</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/graphics.html#geometry">Geometry</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/test.html">Unit and Functional Testing</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/test.html#overview">Overview</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/test.html#handlers">Handlers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/test.html#macros">Macros</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/test.html#functions">Functions</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/test.html#testing-base-julia">Testing Base Julia</a></li>
<li class="toctree-l1"><a class="reference internal" href="../stdlib/profile.html">Profiling</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/profile.html#basic-usage">Basic usage</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/profile.html#accumulation-and-clearing">Accumulation and clearing</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/profile.html#options-for-controlling-the-display-of-profile-results">Options for controlling the display of profile results</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/profile.html#configuration">Configuration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../stdlib/profile.html#function-reference">Function reference</a></li>
</ul>
</li>
</ul>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../devdocs/julia.html">Documentation of Julia’s Internals</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../devdocs/cartesian.html">Base.Cartesian</a></li>
<li class="toctree-l2"><a class="reference internal" href="../devdocs/sysimg.html">System Image Building</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="../devdocs/C.html">Developing/debugging Julia’s C code</a><ul>
<li class="toctree-l2"><a class="reference internal" href="../devdocs/backtraces.html">Reporting and analyzing crashes (segfaults)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../devdocs/debuggingtips.html">gdb debugging tips</a></li>
</ul>
</li>
</ul>
</div>
</nav>
<section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">
<nav class="wy-nav-top" role="navigation" aria-label="top navigation">
<i data-toggle="wy-nav-top" class="fa fa-bars"></i>
<a href="../index.html">Julia Language</a>
</nav>
<div class="wy-nav-content">
<div class="rst-content">
<div role="navigation" aria-label="breadcrumbs navigation">
<ul class="wy-breadcrumbs">
<li><a href="../index.html">Docs</a> »</li>
<li>Linear algebra</li>
<li class="wy-breadcrumbs-aside">
<a href="../_sources/manual/linear-algebra.txt" rel="nofollow"> View page source</a>
</li>
</ul>
<hr/>
</div>
<div role="main" class="document">
<div class="section" id="linear-algebra">
<h1>Linear algebra<a class="headerlink" href="#linear-algebra" title="Permalink to this headline">¶</a></h1>
<div class="section" id="matrix-factorizations">
<h2>Matrix factorizations<a class="headerlink" href="#matrix-factorizations" title="Permalink to this headline">¶</a></h2>
<p><a class="reference external" href="http://en.wikipedia.org/wiki/Matrix_decomposition">Matrix factorizations (a.k.a. matrix decompositions)</a>
compute the factorization of a matrix into a product of matrices, and
are one of the central concepts in linear algebra.</p>
<p>The following table summarizes the types of matrix factorizations that have been
implemented in Julia. Details of their associated methods can be found
in the <a class="reference internal" href="../stdlib/linalg.html#stdlib-linalg"><em>Linear Algebra</em></a> section of the standard library documentation.</p>
<table border="1" class="docutils">
<colgroup>
<col width="15%" />
<col width="85%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Cholesky</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Cholesky factorization</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">CholeskyPivoted</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Pivot_element">Pivoted</a> Cholesky factorization</td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">LU</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/LU_decomposition">LU factorization</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">LUTridiagonal</span></tt></td>
<td>LU factorization for Tridiagonal matrices</td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">UmfpackLU</span></tt></td>
<td>LU factorization for sparse matrices (computed by UMFPack)</td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">QR</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/QR_decomposition">QR factorization</a></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">QRCompactWY</span></tt></td>
<td>Compact WY form of the QR factorization</td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">QRPivoted</span></tt></td>
<td>Pivoted <a class="reference external" href="http://en.wikipedia.org/wiki/QR_decomposition">QR factorization</a></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Hessenberg</span></tt></td>
<td><a class="reference external" href="http://mathworld.wolfram.com/HessenbergDecomposition.html">Hessenberg decomposition</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Eigen</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Eigendecomposition_(matrix)">Spectral decomposition</a></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">SVD</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Singular value decomposition</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">GeneralizedSVD</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Generalized_singular_value_decomposition#Higher_order_version">Generalized SVD</a></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="special-matrices">
<h2>Special matrices<a class="headerlink" href="#special-matrices" title="Permalink to this headline">¶</a></h2>
<p><a class="reference external" href="http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=3274">Matrices with special symmetries and structures</a>
arise often in linear algebra and are frequently associated with
various matrix factorizations.
Julia features a rich collection of special matrix types, which allow for fast
computation with specialized routines that are specially developed for
particular matrix types.</p>
<p>The following tables summarize the types of special matrices that have been
implemented in Julia, as well as whether hooks to various optimized methods
for them in LAPACK are available.</p>
<table border="1" class="docutils">
<colgroup>
<col width="19%" />
<col width="81%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Hermitian</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Hermitian_matrix">Hermitian matrix</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Triangular</span></tt></td>
<td>Upper/lower <a class="reference external" href="http://en.wikipedia.org/wiki/Triangular_matrix">triangular matrix</a></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Tridiagonal</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Tridiagonal_matrix">Tridiagonal matrix</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">SymTridiagonal</span></tt></td>
<td>Symmetric tridiagonal matrix</td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Bidiagonal</span></tt></td>
<td>Upper/lower <a class="reference external" href="http://en.wikipedia.org/wiki/Bidiagonal_matrix">bidiagonal matrix</a></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Diagonal</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Diagonal_matrix">Diagonal matrix</a></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">UniformScaling</span></tt></td>
<td><a class="reference external" href="http://en.wikipedia.org/wiki/Uniform_scaling">Uniform scaling operator</a></td>
</tr>
</tbody>
</table>
<div class="section" id="elementary-operations">
<h3>Elementary operations<a class="headerlink" href="#elementary-operations" title="Permalink to this headline">¶</a></h3>
<table border="1" class="docutils">
<colgroup>
<col width="29%" />
<col width="10%" />
<col width="10%" />
<col width="10%" />
<col width="10%" />
<col width="30%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td>Matrix type</td>
<td><tt class="docutils literal"><span class="pre">+</span></tt></td>
<td><tt class="docutils literal"><span class="pre">-</span></tt></td>
<td><tt class="docutils literal"><span class="pre">*</span></tt></td>
<td><tt class="docutils literal"><span class="pre">\</span></tt></td>
<td>Other functions with
optimized methods</td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Hermitian</span></tt></td>
<td> </td>
<td> </td>
<td> </td>
<td>XY</td>
<td><tt class="docutils literal"><span class="pre">inv</span></tt>,
<tt class="docutils literal"><span class="pre">sqrtm</span></tt>, <tt class="docutils literal"><span class="pre">expm</span></tt></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Triangular</span></tt></td>
<td> </td>
<td> </td>
<td>XY</td>
<td>XY</td>
<td><tt class="docutils literal"><span class="pre">inv</span></tt>, <tt class="docutils literal"><span class="pre">det</span></tt></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">SymTridiagonal</span></tt></td>
<td>X</td>
<td>X</td>
<td>XZ</td>
<td>XY</td>
<td><tt class="docutils literal"><span class="pre">eigmax/min</span></tt></td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Tridiagonal</span></tt></td>
<td>X</td>
<td>X</td>
<td>XZ</td>
<td>XY</td>
<td> </td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Bidiagonal</span></tt></td>
<td>X</td>
<td>X</td>
<td>XZ</td>
<td>XY</td>
<td> </td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Diagonal</span></tt></td>
<td>X</td>
<td>X</td>
<td>XY</td>
<td>XY</td>
<td><tt class="docutils literal"><span class="pre">inv</span></tt>, <tt class="docutils literal"><span class="pre">det</span></tt>,
<tt class="docutils literal"><span class="pre">logdet</span></tt>, <tt class="docutils literal"><span class="pre">/</span></tt></td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">UniformScaling</span></tt></td>
<td>X</td>
<td>X</td>
<td>XYZ</td>
<td>XYZ</td>
<td><tt class="docutils literal"><span class="pre">/</span></tt></td>
</tr>
</tbody>
</table>
<p>Legend:</p>
<table border="1" class="docutils">
<colgroup>
<col width="5%" />
<col width="95%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td>X</td>
<td>An optimized method for matrix-matrix operations is available</td>
</tr>
<tr class="row-even"><td>Y</td>
<td>An optimized method for matrix-vector operations is available</td>
</tr>
<tr class="row-odd"><td>Z</td>
<td>An optimized method for matrix-scalar operations is available</td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="id2">
<h3>Matrix factorizations<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h3>
<table border="1" class="docutils">
<colgroup>
<col width="22%" />
<col width="9%" />
<col width="10%" />
<col width="14%" />
<col width="14%" />
<col width="10%" />
<col width="21%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Matrix type</th>
<th class="head">LAPACK</th>
<th class="head"><tt class="docutils literal"><span class="pre">eig</span></tt></th>
<th class="head"><tt class="docutils literal"><span class="pre">eigvals</span></tt></th>
<th class="head"><tt class="docutils literal"><span class="pre">eigvecs</span></tt></th>
<th class="head"><tt class="docutils literal"><span class="pre">svd</span></tt></th>
<th class="head"><tt class="docutils literal"><span class="pre">svdvals</span></tt></th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Hermitian</span></tt></td>
<td>HE</td>
<td> </td>
<td>ABC</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Triangular</span></tt></td>
<td>TR</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">SymTridiagonal</span></tt></td>
<td>ST</td>
<td>A</td>
<td>ABC</td>
<td>AD</td>
<td> </td>
<td> </td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Tridiagonal</span></tt></td>
<td>GT</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr class="row-even"><td><tt class="docutils literal"><span class="pre">Bidiagonal</span></tt></td>
<td>BD</td>
<td> </td>
<td> </td>
<td> </td>
<td>A</td>
<td>A</td>
</tr>
<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Diagonal</span></tt></td>
<td>DI</td>
<td> </td>
<td>A</td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p>Legend:</p>
<table border="1" class="docutils">
<colgroup>
<col width="2%" />
<col width="83%" />
<col width="15%" />
</colgroup>
<tbody valign="top">
<tr class="row-odd"><td>A</td>
<td>An optimized method to find all the characteristic values and/or vectors is available</td>
<td>e.g. <tt class="docutils literal"><span class="pre">eigvals(M)</span></tt></td>
</tr>
<tr class="row-even"><td>B</td>
<td>An optimized method to find the <tt class="docutils literal"><span class="pre">il</span></tt><sup>th</sup> through the <tt class="docutils literal"><span class="pre">ih</span></tt><sup>th</sup> characteristic values are available</td>
<td><tt class="docutils literal"><span class="pre">eigvals(M,</span> <span class="pre">il,</span> <span class="pre">ih)</span></tt></td>
</tr>
<tr class="row-odd"><td>C</td>
<td>An optimized method to find the characteristic values in the interval [<tt class="docutils literal"><span class="pre">vl</span></tt>, <tt class="docutils literal"><span class="pre">vh</span></tt>] is available</td>
<td><tt class="docutils literal"><span class="pre">eigvals(M,</span> <span class="pre">vl,</span> <span class="pre">vh)</span></tt></td>
</tr>
<tr class="row-even"><td>D</td>
<td>An optimized method to find the characteristic vectors corresponding to the characteristic values <tt class="docutils literal"><span class="pre">x=[x1,</span> <span class="pre">x2,...]</span></tt> is available</td>
<td><tt class="docutils literal"><span class="pre">eigvecs(M,</span> <span class="pre">x)</span></tt></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="the-uniform-scaling-operator">
<h3>The uniform scaling operator<a class="headerlink" href="#the-uniform-scaling-operator" title="Permalink to this headline">¶</a></h3>
<p>A <tt class="docutils literal"><span class="pre">UniformScaling</span></tt> operator represents a scalar times the identity operator, <tt class="docutils literal"><span class="pre">λ*I</span></tt>. The identity operator <tt class="docutils literal"><span class="pre">I</span></tt> is defined as a constant and is an instance of <tt class="docutils literal"><span class="pre">UniformScaling</span></tt>. The size of these operators are generic and match the other matrix in the binary operations <tt class="docutils literal"><span class="pre">+</span></tt>,``-<tt class="docutils literal"><span class="pre">,``*</span></tt> and <tt class="docutils literal"><span class="pre">\</span></tt>. For <tt class="docutils literal"><span class="pre">A+I</span></tt> and <tt class="docutils literal"><span class="pre">A-I</span></tt> this means that <tt class="docutils literal"><span class="pre">A</span></tt> must be square. Multiplication with the identity operator <tt class="docutils literal"><span class="pre">I</span></tt> is a noop (except for checking that the scaling factor is one) and therefore almost without overhead.</p>
</div>
</div>
</div>
</div>
<footer>
<div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
<a href="networking-and-streams.html" class="btn btn-neutral float-right" title="Networking and Streams"/>Next <span class="fa fa-arrow-circle-right"></span></a>
<a href="arrays.html" class="btn btn-neutral" title="Multi-dimensional Arrays"><span class="fa fa-arrow-circle-left"></span> Previous</a>
</div>
<hr/>
<div role="contentinfo">
<p>
</p>
</div>
<a href="https://github.com/snide/sphinx_rtd_theme">Sphinx theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>
</footer>
</div>
</div>
</section>
</div>
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT:'../',
VERSION:'0.3.4',
COLLAPSE_INDEX:false,
FILE_SUFFIX:'.html',
HAS_SOURCE: true
};
</script>
<script type="text/javascript" src="../_static/jquery.js"></script>
<script type="text/javascript" src="../_static/underscore.js"></script>
<script type="text/javascript" src="../_static/doctools.js"></script>
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript" src="../_static/js/theme.js"></script>
<script type="text/javascript">
jQuery(function () {
SphinxRtdTheme.StickyNav.enable();
});
</script>
</body>
</html>
