{"id":5303,"date":"2014-04-27T06:05:18","date_gmt":"2014-04-26T21:05:18","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5303"},"modified":"2014-08-01T18:26:38","modified_gmt":"2014-08-01T09:26:38","slug":"theorems-on-eigenvalues-and-eigenvectors","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5303","title":{"rendered":"Theorems on eigenvalues and eigenvectors"},"content":{"rendered":"
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Theorem 12. <\/strong> The eigenvalues of a Hermitian matrix [or symmetric real matrix] are real. The eigenvalues of a skew-Hermitian matrix [or skew-symmetric real matrix] are zero or pure imaginary. The eigenvalues of a unitary [or real orthogonal matrix] all have absolute value equal to 1. <\/p>\n

Theorem 13. <\/strong> The eigenvectors belonging to different eigenvalues of a Hermitian matrix [or symmetric real matrix] are orthogonal. <\/p>\n

Theorem 14. [Cayley-Hamilton]<\/strong> A matrix satisfies its own characteristic equation. <\/p>\n

Theorem 15. [reduction of matrix to diagonal form]<\/strong> If a non-singular matrix A has distinct eigenvalues \\lambda_1,\\ \\lambda_2,\\ \\cdots with corresponding eigenvectors written as columns in the matrix <\/p>\n

\\displaystyle B = \\left( \\begin{array}{cccc} b_{11} & b_{12} & b_{13} & \\cdots \\\\ b_{21} & b_{22} & b_{23} & \\cdots \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\end{array} \\right)<\/p>\n

then <\/p>\n

B^{-1}AB = \\left( \\begin{array}{cccc} \\lambda_1 & 0 & 0 & \\cdots \\\\ 0 & \\lambda_2 & 0 & \\cdots \\\\ 0 & 0 & \\lambda_3 & \\cdots \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\end{array} \\right)<\/p>\n

i.e. B^{-1}AB, called the transform<\/em> of A by B, is a diagonal matrix containing the eigenvalues of A in the main diagonal and zeros elsewhere. We say that A has been transformed<\/em> or reduced to diagonal form<\/em>. <\/p>\n

Theorem 16. [Reduction of quadratic form to canonical form]<\/strong> Let A be a symmetric matrix, for example, <\/p>\n

\\displaystyle A = \\left( \\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\end{array} \\right),\\ a_{12} = a_{21},\\ a_{13} = a_{11},\\ a_{23} = a_{32} <\/p>\n

Then if \\displaystyle X = \\left( \\begin{array}{c} x_1 \\\\ x_2 \\\\ x_3 \\end{array} \\right), we obtain the quadratic form<\/em> <\/p>\n

X^TAX = a_{11}x_1^2 + a_{22}x_2^2 + a_{33}x_3^2 + 2a_{12}x_1x_2 + 2a_{13}x_1x_3 + 2a_{23}x_2x_3<\/p>\n

The cross product terms of this quadratic form can be removed by letting X = BU where U is the column vector with elements u_1,\\ u_2,\\ u_3 and B is an orthogonal matrix which diagonalizes A. The new quadratic form in u_1,\\ u_2,\\ u_3 with no cross product terms is called the canonical form<\/em>. A generalization can be made to Hermitian quadratic forms. <\/p>\n<\/blockquote>\n