{"id":33868,"date":"2022-04-24T20:43:18","date_gmt":"2022-04-25T00:43:18","guid":{"rendered":"https:\/\/nstem.org\/staging\/?p=33868"},"modified":"2022-04-25T17:33:08","modified_gmt":"2022-04-25T21:33:08","slug":"an-algebraic-and-geometric-approach-to-eigenvalues-and-eigenvectors","status":"publish","type":"post","link":"https:\/\/nstem.org\/staging\/2022\/04\/an-algebraic-and-geometric-approach-to-eigenvalues-and-eigenvectors\/","title":{"rendered":"An Algebraic and Geometric Approach to Eigenvalues and Eigenvectors"},"content":{"rendered":"

In linear algebra, we are introduced to eigenvectors and eigenvalues. While eigenvectors are vectors with a direction immutable by a transformation, eigenvalues are associated with the physical quantity by which the eigenvector is scaled.\u00a0<\/span><\/p>\n

For the average student, most of these concepts can be difficult to grasp from a geometric perspective. That is why, in the following sections, I will utilize visual representations in addition to manipulating symbols algebraically.\u00a0<\/span><\/p>\n

The fundamental formula that characterizes the eigenvectors and eigenvalues of a matrix is Av = \u03bbv. In which <\/span>A<\/span><\/i> is a matrix, <\/span>v<\/span><\/i> is an eigenvector, <\/span>\u03bb<\/span><\/i> (lambda) is an eigenvalue, and <\/span>v<\/span><\/i> is an eigenvector.<\/span><\/p>\n

Let\u2019s begin by considering the given <\/span>2×2<\/span><\/i> matrix.<\/span><\/p>\n

\u00a0[ 2, 3<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05, 4 ]<\/span><\/p>\n

Where our eigenvectors are:<\/span><\/p>\n

[ 3\u00a0 \u00a0 \u00a0 and\u00a0 \u00a0 \u00a0 [ -1<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a05 ] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1 ]\u00a0<\/span><\/p>\n

With corresponding eigenvalues of <\/span>\u03bb<\/span><\/i>1<\/span><\/i> = 7 <\/span><\/i>and<\/span> \u03bb<\/span><\/i>2<\/span><\/i> = -1.<\/span><\/i><\/p>\n

To arrive at these conclusions, our first step consists of subtracting \u03bb from the diagonal entries, much like this:<\/span><\/p>\n

[ 2 – \u03bb, 3<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05, 4 – \u03bb\u00a0 ]<\/span><\/p>\n

Wherefore the characteristic equation, or determinant, of our matrix, is:<\/span><\/p>\n

(\u03bb – 7) (\u03bb +1) = 0<\/span><\/p>\n

We end up with <\/span>\u03bb<\/span><\/i>1<\/span><\/i> = 7<\/span><\/i> and <\/span>\u03bb<\/span><\/i>2<\/span><\/i> = -1<\/span><\/i> as our eigenvalues. Now, we will input the different values of \u03bb into our previous equation.<\/span><\/p>\n

[ 2 – 7, 3 \u00a0 \u00a0 =\u00a0 \u00a0 [ -5, 3\u00a0<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a05, 4 – 7\u00a0 ] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 5, -3 ]<\/span><\/p>\n

And<\/span><\/p>\n

[ 2 – -1, 3 \u00a0 \u00a0 =\u00a0 \u00a0 [ 3, 3\u00a0<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a05, 4 – -1\u00a0 ]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 5, 5 ]<\/span><\/p>\n

Where the null spaces of our matrix, found by satisfying the equation <\/span>Ax = 0<\/span><\/i>, are:<\/span><\/p>\n

\u03bb<\/span>1<\/span> = {[ 3<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a05 ]}<\/span><\/p>\n

And<\/span><\/p>\n

\u03bb<\/span>2<\/span> = {[ -1<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 ]}<\/span><\/p>\n

This is how we reach the aforementioned solutions. On another note, a visual that can accurately represent our conclusion would be the following:<\/span><\/p>\n

\"\"<\/p>\n

Image source<\/a><\/p>\n

Hopefully, with this brief overview, you\u2019ve been able to gain some confidence in your ability to calculate and visualize eigenvectors and eigenvalues from an algebraic and geometric point of view.<\/span><\/p>\n

By: Estefania Olaiz<\/strong><\/em><\/p>\n

March 19, 2022<\/strong><\/em><\/p>\n

Sources:<\/b><\/p>\n

    \n
  1. https:\/\/en.wikipedia.org\/wiki\/Eigenvalues_and_eigenvectors#:~:text=Geometrically%2C%20an%20eigenvector%2C%20corresponding%20to,negative%2C%20the%20direction%20is%20reversed<\/span><\/a>.<\/span><\/li>\n
  2. https:\/\/www.mathsisfun.com\/algebra\/eigenvalue.html<\/span><\/a>\u00a0<\/span><\/li>\n
  3. https:\/\/lpsa.swarthmore.edu\/MtrxVibe\/EigMat\/MatrixEigen.html<\/span><\/a><\/li>\n
  4. https:\/\/www.mathsisfun.com\/algebra\/matrix-determinant.html<\/span><\/a>\u00a0<\/span><\/li>\n
  5. https:\/\/www.geogebra.org\/m\/JP2XZpzV<\/span><\/a><\/li>\n<\/ol>\n

     <\/p>\n

     <\/p>\n","protected":false},"excerpt":{"rendered":"

    In linear algebra, we are introduced to eigenvectors and eigenvalues. While eigenvectors are vectors with a direction immutable by a transformation, eigenvalues are associated with the physical quantity by which the eigenvector is scaled.\u00a0 For the average student, most of these concepts can be difficult to grasp from a geometric perspective. That is why, in […]<\/p>\n","protected":false},"author":9,"featured_media":33869,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pmpro_default_level":"","_kad_blocks_custom_css":"","_kad_blocks_head_custom_js":"","_kad_blocks_body_custom_js":"","_kad_blocks_footer_custom_js":"","_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[198],"tags":[606,603,604,605,607,608,602,609,457,213],"persona":[],"class_list":{"0":"post-33868","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-blog","8":"tag-algebra","9":"tag-eigenvalues","10":"tag-eigenvectors","11":"tag-geometry","12":"tag-linear-algebra","13":"tag-matrices","14":"tag-university","15":"tag-vectors","16":"tag-college","17":"tag-mathematics","18":"pmpro-has-access","19":"entry"},"yoast_head":"\nAn Algebraic and Geometric Approach to Eigenvalues and Eigenvectors National STEM\u2122 Honor Society<\/title>\n<meta name=\"description\" content=\"In linear algebra, we are introduced to eigenvectors and eigenvalues. 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