{"id":687301,"date":"2023-09-13T18:06:23","date_gmt":"2023-09-13T12:36:23","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=687301"},"modified":"2023-09-13T18:09:08","modified_gmt":"2023-09-13T12:39:08","slug":"eigen-values-2","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/","title":{"rendered":"Eigen Values"},"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\/articles\/eigen-values\/#Introduction_to_Eigen_Values\" title=\"Introduction to Eigen Values\">Introduction to Eigen Values<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Definition_of_Eigen_values_and_Vector_of_a_Matrix\" title=\"Definition of Eigen values and Vector of a Matrix\">Definition of Eigen values and Vector of a Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Eigen_Values_and_Vector_Equation\" title=\"Eigen Values and Vector Equation:\">Eigen Values and Vector Equation:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Eigen_Vector_Method\" title=\"Eigen Vector Method\">Eigen Vector Method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#How_to_Find_Eigen_Values_and_Eigen_Vector\" title=\"How to Find Eigen Values and Eigen Vector\">How to Find Eigen Values and Eigen Vector<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Applications_of_Eigen_Values_and_Vectors\" title=\"Applications of Eigen Values and Vectors\">Applications of Eigen Values and Vectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Solved_examples_on_eigen_values_and_Eigen_vectors\" title=\"Solved examples on eigen values and Eigen vectors\">Solved examples on eigen values and Eigen vectors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Conclusion\" title=\"Conclusion\">Conclusion<\/a><\/li><\/ul><\/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\/articles\/eigen-values\/#Frequently_Asked_Questions_on_Eigen_values_and_Eigen_vectors\" title=\"Frequently Asked Questions on Eigen values and Eigen vectors\">Frequently Asked Questions on Eigen values and Eigen vectors<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Can_a_matrix_have_no_eigen_vectors\" title=\"Can a matrix have no eigen vectors? \">Can a matrix have no eigen vectors? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Do_all_eigen_vectors_have_corresponding_eigenvalues\" title=\"Do all eigen vectors have corresponding eigenvalues? \">Do all eigen vectors have corresponding eigenvalues? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#How_are_eigen_vectors_used_in_data_analysis\" title=\"How are eigen vectors used in data analysis? \">How are eigen vectors used in data analysis? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Are_eigen_vectors_unique_for_a_matrix\" title=\"Are eigen vectors unique for a matrix? \">Are eigen vectors unique for a matrix? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Can_a_matrix_have_repeated_eigenvalues\" title=\"Can a matrix have repeated eigenvalues? \">Can a matrix have repeated eigenvalues? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#What_is_eigen_value_and_eigen_vector_of_a_matrix\" title=\"What is eigen value and eigen vector of a matrix?\">What is eigen value and eigen vector of a matrix?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#Can_a_2_by_2_matrix_have_eigen_vectors\" title=\"Can a 2 by 2 matrix have eigen vectors? \">Can a 2 by 2 matrix have eigen vectors? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#What_is_the_formula_for_eigen_vectros\" title=\"What is the formula for eigen vectros? \">What is the formula for eigen vectros? <\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Eigen_Values\"><\/span>Introduction to Eigen Values<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Eigen vectors<\/strong> are fundamental concepts in linear algebra that play a crucial role in various fields, from physics to computer science. They provide insights into the behavior of linear transformations encoded in matrices. This article delves into the definition, equations, methods, and examples of eigen vectors.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Definition_of_Eigen_values_and_Vector_of_a_Matrix\"><\/span>Definition of Eigen values and Vector of a Matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>An eigen vector of a square matrix represents a non-zero vector that, when multiplied by the matrix, results in a scaled version of itself. In simpler terms, the direction of the <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/vectors\/\"><strong>vector<\/strong> <\/a>remains unchanged, though its length may change.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Eigen_Values_and_Vector_Equation\"><\/span>Eigen Values and Vector Equation:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>For a matrix A and an eigen vector v, the eigen vector equation is given by: A****v = \u03bbv<\/p>\n<p>Here, A is the matrix, v is the eigen vector, and \u03bb (lambda) is the scalar value known as the eigenvalue corresponding to v.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Eigen_Vector_Method\"><\/span>Eigen Vector Method<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The eigen vector method involves solving the eigen vector equation to find both eigenvalues and corresponding eigen vectors. It is often accomplished using techniques like the characteristic polynomial, eigenvalue decomposition, or iterative algorithms.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"How_to_Find_Eigen_Values_and_Eigen_Vector\"><\/span>How to Find Eigen Values and Eigen Vector<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>To find an eigen vector for a given eigenvalue \u03bb, we substitute \u03bb into the equation (A &#8211; \u03bbI)v = 0, where I is the identity matrix. The non-trivial solution of this equation gives the eigen vector.<\/p>\n<p>Eigen vectors can have both left and right forms. Left eigen vectors result from the equation u****A = \u03bbu, where u is the left eigen vector. Right eigen vectors result from the equation A****v = \u03bbv, where v is the right eigen vector. The left and right eigen vectors can be different for non-symmetric matrices.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Applications_of_Eigen_Values_and_Vectors\"><\/span>Applications of Eigen Values and Vectors<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Eigen vectors of a matrix have diverse applications across various fields due to their ability to capture essential properties of linear transformations. These applications include:<\/p>\n<ul>\n<li><strong>Quantum Mechanics:<\/strong> Eigen vectors represent stationary states in particle behavior.<\/li>\n<li><strong>Data Analysis:<\/strong> Principal Component Analysis (PCA) reduces data dimensions while retaining patterns.<\/li>\n<li><strong>Structural Engineering:<\/strong> Identifies modes of vibration and stability in structures.<\/li>\n<li><strong>Image Compression:<\/strong> Efficiently compresses images by retaining crucial information.<\/li>\n<li><strong>Mechanical Engineering:<\/strong> Analyzes vibrations and stresses in mechanical systems.<\/li>\n<li><strong>Finance:<\/strong> Used in portfolio optimization and risk management.<\/li>\n<li><strong>Computer Graphics:<\/strong> Shapes analysis, animation, and image manipulation.<\/li>\n<li><strong>Chemistry:<\/strong> Deciphers molecular vibrational modes in spectroscopy.<\/li>\n<li><strong>Social Networks:<\/strong> Identifies influential nodes and communities.<\/li>\n<li><strong>Signal Processing:<\/strong> Aids in noise reduction, speech recognition, and image filtering.<\/li>\n<\/ul>\n<p>In various domains, eigen vectors enhance understanding and problem-solving capabilities.<\/p>\n<p><strong>Also Check For:<\/strong><\/p>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/articles\/math-articles\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Articles<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/formulas\/math-formulas\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Formulas<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/matrix-multiplication\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Matrix Multiplication<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Identity Matrix<\/button><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Solved_examples_on_eigen_values_and_Eigen_vectors\"><\/span>Solved examples on eigen values and Eigen vectors<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Example 1.<\/strong> Find the eigen values and eigen vectors of matrix A = [2 2; 1 3]\n<p>Solution: Eigen values = 4, 1; Eigen vectors = [1; 1], [-2; 1].<\/p>\n<p><strong>Example 2.<\/strong> For matrix B = [3 2; 1 4], compute the left and right eigen vectors.<\/p>\n<p>Solution: Left eigen vector = [2; -1], Right eigen vector = [1; 1].<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Conclusion\"><\/span>Conclusion<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Eigen vectors offer profound insights into the behavior of linear transformations encapsulated in matrices. They find applications in various fields, including quantum mechanics, data analysis, and computer graphics, enhancing our understanding of complex systems.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Frequently_Asked_Questions_on_Eigen_values_and_Eigen_vectors\"><\/span>Frequently Asked Questions on Eigen values and Eigen vectors<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Can_a_matrix_have_no_eigen_vectors\"><\/span>Can a matrix have no eigen vectors? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, some matrices do not have eigen vectors, which typically occurs in complex or irregular matrices. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Do_all_eigen_vectors_have_corresponding_eigenvalues\"><\/span>Do all eigen vectors have corresponding eigenvalues? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, each eigen vector corresponds to a specific eigenvalue. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_are_eigen_vectors_used_in_data_analysis\"><\/span>How are eigen vectors used in data analysis? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tEigen vectors are used in techniques like Principal Component Analysis (PCA) to reduce data dimensions while preserving meaningful information. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Are_eigen_vectors_unique_for_a_matrix\"><\/span>Are eigen vectors unique for a matrix? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tEigen vectors are unique up to a scalar multiple. In other words, any scalar multiple of an eigen vector is also an eigen vector. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Can_a_matrix_have_repeated_eigenvalues\"><\/span>Can a matrix have repeated eigenvalues? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, matrices can have repeated eigenvalues, and this often leads to more than one linearly independent eigen vector for the same eigenvalue. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_eigen_value_and_eigen_vector_of_a_matrix\"><\/span>What is eigen value and eigen vector of a matrix?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tEigenvalues and eigenvectors of a matrix are fundamental concepts in linear algebra. Eigenvalues are scalar values that scale eigenvectors, which are non-zero vectors representing directions unaffected by matrix transformations. They play a crucial role in various applications, from quantum mechanics to data analysis. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Can_a_2_by_2_matrix_have_eigen_vectors\"><\/span>Can a 2 by 2 matrix have eigen vectors? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, a 2 by 2 matrix can have eigen vectors. Eigen vectors and eigen values are concepts applicable to square matrices, including 2 by 2 matrices. For a 2 by 2 matrix, you can find its eigenvalues and corresponding eigenvectors using algebraic methods or computational techniques. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_formula_for_eigen_vectros\"><\/span>What is the formula for eigen vectros? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe formula to find eigen vectors for a square matrix A and its eigenvalue \u03bb is: (A - \u03bbI*)v = 0 Here, I is the identity matrix, v is the eigen vector, and 0 is the zero vector. Solving this equation yields the eigen vector corresponding to the given eigenvalue. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Can a matrix have no eigen vectors? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, some matrices do not have eigen vectors, which typically occurs in complex or irregular matrices.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Do all eigen vectors have corresponding eigenvalues? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, each eigen vector corresponds to a specific eigenvalue.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How are eigen vectors used in data analysis? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Eigen vectors are used in techniques like Principal Component Analysis (PCA) to reduce data dimensions while preserving meaningful information.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Are eigen vectors unique for a matrix? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Eigen vectors are unique up to a scalar multiple. In other words, any scalar multiple of an eigen vector is also an eigen vector.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Can a matrix have repeated eigenvalues? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, matrices can have repeated eigenvalues, and this often leads to more than one linearly independent eigen vector for the same eigenvalue.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is eigen value and eigen vector of a matrix?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Eigenvalues and eigenvectors of a matrix are fundamental concepts in linear algebra. Eigenvalues are scalar values that scale eigenvectors, which are non-zero vectors representing directions unaffected by matrix transformations. They play a crucial role in various applications, from quantum mechanics to data analysis.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Can a 2 by 2 matrix have eigen vectors? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, a 2 by 2 matrix can have eigen vectors. Eigen vectors and eigen values are concepts applicable to square matrices, including 2 by 2 matrices. For a 2 by 2 matrix, you can find its eigenvalues and corresponding eigenvectors using algebraic methods or computational techniques.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the formula for eigen vectros? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The formula to find eigen vectors for a square matrix A and its eigenvalue \u03bb is: (A - \u03bbI*)v = 0 Here, I is the identity matrix, v is the eigen vector, and 0 is the zero vector. Solving this equation yields the eigen vector corresponding to the given eigenvalue.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Eigen Values Eigen vectors are fundamental concepts in linear algebra that play a crucial role in various fields, [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Eigen Values","_yoast_wpseo_title":"Eigenvalues & Eigenvectors - Definition, Equation, Applications and Examples","_yoast_wpseo_metadesc":"When a matrix acts on a vector, the vector might stretch or shrink without changing its direction. That special vector is called an eigenvector. The amount it stretches or shrinks is the eigenvalue.","custom_permalink":"articles\/eigen-values\/"},"categories":[8442,8443],"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>Eigenvalues &amp; Eigenvectors - Definition, Equation, Applications and Examples<\/title>\n<meta name=\"description\" content=\"When a matrix acts on a vector, the vector might stretch or shrink without changing its direction. That special vector is called an eigenvector. The amount it stretches or shrinks is the eigenvalue.\" \/>\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\/articles\/eigen-values\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Eigenvalues &amp; Eigenvectors - Definition, Equation, Applications and Examples\" \/>\n<meta property=\"og:description\" content=\"When a matrix acts on a vector, the vector might stretch or shrink without changing its direction. That special vector is called an eigenvector. The amount it stretches or shrinks is the eigenvalue.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/\" \/>\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=\"2023-09-13T12:36:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-09-13T12:39:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1920\" \/>\n\t<meta property=\"og:image:height\" content=\"1008\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\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=\"Ankit\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Eigenvalues & Eigenvectors - Definition, Equation, Applications and Examples","description":"When a matrix acts on a vector, the vector might stretch or shrink without changing its direction. That special vector is called an eigenvector. The amount it stretches or shrinks is the eigenvalue.","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\/articles\/eigen-values\/","og_locale":"en_US","og_type":"article","og_title":"Eigenvalues & Eigenvectors - Definition, Equation, Applications and Examples","og_description":"When a matrix acts on a vector, the vector might stretch or shrink without changing its direction. That special vector is called an eigenvector. The amount it stretches or shrinks is the eigenvalue.","og_url":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_published_time":"2023-09-13T12:36:23+00:00","article_modified_time":"2023-09-13T12:39:08+00:00","og_image":[{"width":1920,"height":1008,"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2025\/04\/infinitylearn.jpg","type":"image\/jpeg"}],"twitter_card":"summary_large_image","twitter_creator":"@InfinityLearn_","twitter_site":"@InfinityLearn_","twitter_misc":{"Written by":"Ankit","Est. reading time":"4 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":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/","name":"Eigenvalues & Eigenvectors - Definition, Equation, Applications and Examples","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"datePublished":"2023-09-13T12:36:23+00:00","dateModified":"2023-09-13T12:39:08+00:00","description":"When a matrix acts on a vector, the vector might stretch or shrink without changing its direction. That special vector is called an eigenvector. The amount it stretches or shrinks is the eigenvalue.","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Eigen Values"}]},{"@type":"Article","@id":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#article","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#webpage"},"author":{"@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d647d4ff3a1111ff8eeccdb6b12651cb"},"headline":"Eigen Values","datePublished":"2023-09-13T12:36:23+00:00","dateModified":"2023-09-13T12:39:08+00:00","mainEntityOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/articles\/eigen-values\/#webpage"},"wordCount":839,"publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"articleSection":["Articles","Math Articles"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d647d4ff3a1111ff8eeccdb6b12651cb","name":"Ankit","image":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/b1068bdc2711bd9c9f8be3b229f758f6?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/b1068bdc2711bd9c9f8be3b229f758f6?s=96&d=mm&r=g","caption":"Ankit"},"url":"https:\/\/infinitylearn.com\/surge\/author\/ankit\/"}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/687301"}],"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\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/comments?post=687301"}],"version-history":[{"count":0,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/687301\/revisions"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=687301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=687301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/tags?post=687301"},{"taxonomy":"table_tags","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/table_tags?post=687301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}