{"id":5292,"date":"2014-04-26T06:05:45","date_gmt":"2014-04-25T21:05:45","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5292"},"modified":"2014-08-01T18:27:24","modified_gmt":"2014-08-01T09:27:24","slug":"eigenvalues-and-eigenvectors","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5292","title":{"rendered":"Eigenvalues and eigenvectors"},"content":{"rendered":"
\n

Let A = (a_{jk}) be an n \\times n matrix and X a column vector. The equation <\/p>\n

AX = \\lambda X \\cdots (21)<\/p>\n

where \\lambda is a number can be written as <\/p>\n

\\displaystyle \\left( \\begin{array}{cccc} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{array} \\right) \\left( \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\cdots \\\\ x_n \\end{array} \\right) = \\lambda\\left( \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\cdots \\\\ x_n \\end{array} \\right) \\cdots (22)<\/p>\n

or<\/p>\n

\\displaystyle \\left. \\begin{array}{ccc} (a_{11} - \\lambda)x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n & = & 0 \\\\ a_{21}x_1 + (a_{22} - \\lambda)x_2 + \\cdots + a_{2n}x_n & = & 0 \\\\ \\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots & = & 0 \\\\ a_{n1}x_1 + a_{n2}x_2 + \\cdots + (a_{nn} - \\lambda)x_n & = & 0 \\end{array}\\right\\} \\cdots(23)<\/p>\n

The equation (23) will have non-trivial solution if and only if <\/p>\n

\\displaystyle \\left| \\begin{array}{cccc} a_{11} - \\lambda & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} - \\lambda & \\cdots & a_{2n} \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} - \\lambda \\end{array} \\right| = 0 \\cdots(24)<\/p>\n

which is a polynomial equation of degree n<\/em> in \\lambda. The roots of this equation are called eigenvalues<\/em> or characteristic values<\/em> of the matrix A. Corresponding to each eigenvalue there will be a solution X \\ne 0, i.e. a non-trivial solution, which is called an eigenvector<\/em> or characteristic vector<\/em> belonging to the eigenvalue. The equation (24) can also be written <\/p>\n

\\det(A - \\lambda I) = 0 \\cdots(25)<\/p>\n

and the equation in \\lambda is often called the characteristic equation<\/em>. <\/p>\n<\/blockquote>\n