{"id":153540,"date":"2022-03-25T22:32:13","date_gmt":"2022-03-25T17:02:13","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/eigenvector-of-matrix-explanation-example-characteristics-and-applications\/"},"modified":"2023-01-11T10:26:14","modified_gmt":"2023-01-11T04:56:14","slug":"eigenvector-of-matrix-explanation-example-characteristics-and-applications","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/","title":{"rendered":"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Define_Eigenvector\" title=\"Define Eigenvector\">Define Eigenvector<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#To_Find_Eigenvectors_of_Matrix\" title=\"To Find Eigenvectors of Matrix\">To Find Eigenvectors of Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Finding_Eigenvectors\" title=\"Finding Eigenvectors\">Finding Eigenvectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Example_of_Using_an_Eigenvalue\" title=\"Example of Using an Eigenvalue\">Example of Using an Eigenvalue<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#What_Does_The_Eigenvector_of_The_Matrix_Mean\" title=\"What Does The Eigenvector of The Matrix Mean?\">What Does The Eigenvector of The Matrix Mean?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Example_of_Calculating_The_Eigenvector_of_a_Matrix\" title=\"Example of Calculating The Eigenvector of a Matrix\">Example of Calculating The Eigenvector of a Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Characteristics_of_Eigenvalues\" title=\"Characteristics of Eigenvalues\">Characteristics of Eigenvalues<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Orthogonality_of_an_Eigenvector\" title=\"Orthogonality of an Eigenvector\">Orthogonality of an Eigenvector<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#Applications_of_an_Eigenvector\" title=\"Applications of an Eigenvector\">Applications of an Eigenvector<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Define_Eigenvector\"><\/span>Define Eigenvector<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Eigenvector is a linear transformation that preserves the direction of a vector, while reducing its magnitude. It is a special type of matrix that has an eigenvalue associated with it. Eigenvector is used in many areas of mathematics and physics, including quantum mechanics, signal processing, and control theory.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-153539 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications-300x212.jpg\" alt=\"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applica\" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications-300x212.jpg?v=1648227730 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications.jpg?v=1648227730 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"To_Find_Eigenvectors_of_Matrix\"><\/span>To Find Eigenvectors of Matrix<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The eigenvector of a matrix is a vector that is associated with a particular eigenvalue and that doesn&#8217;t change when the matrix is multiplied by a constant. To find the eigenvectors of a matrix, you first need to find the eigenvalues. Once you have the eigenvalues, you can use them to calculate the eigenvectors.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Finding_Eigenvectors\"><\/span>Finding Eigenvectors<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>An eigenvector is a vector that is associated with a particular eigenvalue and remains unchanged under a given transformation. To find the eigenvector associated with a particular eigenvalue, one first finds the eigenvalue of the matrix. Once the eigenvalue is found, the eigenvector can be found by solving the equation:<\/p>\n<p>x&#8217; = lambda x<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Example_of_Using_an_Eigenvalue\"><\/span>Example of Using an Eigenvalue<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>An eigenvalue is used to solve a linear equation. The equation is set up with a matrix and the eigenvalue is used to find the value that will make the matrix determinant zero.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_Does_The_Eigenvector_of_The_Matrix_Mean\"><\/span>What Does The Eigenvector of The Matrix Mean?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The eigenvector of the matrix is a vector that is associated with a particular eigenvalue. The eigenvector is used to calculate the eigenvalue and is used to calculate the characteristic polynomial of the matrix.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Example_of_Calculating_The_Eigenvector_of_a_Matrix\"><\/span>Example of Calculating The Eigenvector of a Matrix<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The eigenvector of a matrix is a vector that is associated with a particular eigenvalue and that has the property that the matrix multiplied by the eigenvector produces the same eigenvalue.<\/p>\n<p>The eigenvector of a matrix A can be calculated using the following steps:<\/p>\n<p>1. Find the eigenvalues of A.<\/p>\n<p>2. Find a vector v that is associated with each eigenvalue.<\/p>\n<p>3. Multiply A by each of the vectors v to produce the eigenvalues.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Characteristics_of_Eigenvalues\"><\/span>Characteristics of Eigenvalues<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The eigenvalues of a matrix are the roots of its characteristic equation.<\/p>\n<p>The eigenvalues are always real numbers.<\/p>\n<p>The eigenvalues are always distinct.<\/p>\n<p>The eigenvalues of a matrix determine its shape and behavior.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Orthogonality_of_an_Eigenvector\"><\/span>Orthogonality of an Eigenvector<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The orthogonality of an eigenvector is a measure of how well the vector is aligned with the coordinate axes. If the eigenvector is perfectly aligned with the coordinate axes, it is said to be orthogonal. A vector that is not orthogonal is said to be non-orthogonal.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Applications_of_an_Eigenvector\"><\/span>Applications of an Eigenvector<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>One application of an eigenvector is in the field of machine learning. In particular, eigenvectors can be used for dimensionality reduction, which is the process of reducing the number of dimensions in a dataset while preserving as much of the original information as possible. This is often done in order to make datasets more manageable and easier to work with.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Define Eigenvector Eigenvector is a linear transformation that preserves the direction of a vector, while reducing its magnitude. It is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Eigenvector of Matrix","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about eigenvector of matrix topic of Eigenvector of Matrix maths in details explained by subject experts on infinitylearn.com.","custom_permalink":"maths\/eigenvector-of-matrix\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn about eigenvector of matrix topic of Eigenvector of Matrix maths in details explained by subject experts on infinitylearn.com.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications - Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"og:description\" content=\"Learn about eigenvector of matrix topic of Eigenvector of Matrix maths in details explained by subject experts on infinitylearn.com.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-25T17:02:13+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-01-11T04:56:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications-300x212.jpg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"vipin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications - Infinity Learn by Sri Chaitanya","description":"Learn about eigenvector of matrix topic of Eigenvector of Matrix maths in details explained by subject experts on infinitylearn.com.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/","og_locale":"en_US","og_type":"article","og_title":"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications - Infinity Learn by Sri Chaitanya","og_description":"Learn about eigenvector of matrix topic of Eigenvector of Matrix maths in details explained by subject experts on infinitylearn.com.","og_url":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_published_time":"2022-03-25T17:02:13+00:00","article_modified_time":"2023-01-11T04:56:14+00:00","og_image":[{"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications-300x212.jpg"}],"twitter_card":"summary_large_image","twitter_creator":"@InfinityLearn_","twitter_site":"@InfinityLearn_","twitter_misc":{"Written by":"vipin","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/infinitylearn.com\/surge\/#organization","name":"Infinity Learn","url":"https:\/\/infinitylearn.com\/surge\/","sameAs":["https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","https:\/\/www.instagram.com\/infinitylearn_by_srichaitanya\/","https:\/\/www.linkedin.com\/company\/infinity-learn-by-sri-chaitanya\/","https:\/\/www.youtube.com\/c\/InfinityLearnEdu","https:\/\/twitter.com\/InfinityLearn_"],"logo":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#logo","inLanguage":"en-US","url":"","contentUrl":"","caption":"Infinity Learn"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/#logo"}},{"@type":"WebSite","@id":"https:\/\/infinitylearn.com\/surge\/#website","url":"https:\/\/infinitylearn.com\/surge\/","name":"Infinity Learn by Sri Chaitanya","description":"Surge","publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/infinitylearn.com\/surge\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#primaryimage","inLanguage":"en-US","url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications.jpg?v=1648227730","contentUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications.jpg?v=1648227730","width":606,"height":428},{"@type":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/","name":"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications - Infinity Learn by Sri Chaitanya","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"primaryImageOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#primaryimage"},"datePublished":"2022-03-25T17:02:13+00:00","dateModified":"2023-01-11T04:56:14+00:00","description":"Learn about eigenvector of matrix topic of Eigenvector of Matrix maths in details explained by subject experts on infinitylearn.com.","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications"}]},{"@type":"Article","@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#article","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#webpage"},"author":{"@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2"},"headline":"Eigenvector of Matrix \u2013 Explanation, Example, Characteristics and Applications","datePublished":"2022-03-25T17:02:13+00:00","dateModified":"2023-01-11T04:56:14+00:00","mainEntityOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#webpage"},"wordCount":481,"publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/eigenvector-of-matrix\/#primaryimage"},"thumbnailUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/eigenvector-of-matrix-explanation-example-characteristics-and-applications-300x212.jpg","articleSection":["Maths"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2","name":"vipin","image":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","caption":"vipin"},"sameAs":["http:\/\/surge.infinitylearn.com"],"url":"https:\/\/infinitylearn.com\/surge\/author\/vipin\/"}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/153540"}],"collection":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/comments?post=153540"}],"version-history":[{"count":0,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/153540\/revisions"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=153540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=153540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/tags?post=153540"},{"taxonomy":"table_tags","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/table_tags?post=153540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}