“Symmetric Matrix”的概念、定义、习题解题方法及翻译-数学辞典

单词

Symmetric Matrix

释义

Symmetric Matrix

A symmetric matrix is a Square Matrix which satisfies where denotes theTranspose, so . This also implies

(1)

where I is the Identity Matrix. Written explicitly,

(2)

The symmetric part of any Matrix may be obtained from

(3)

A Matrix A is symmetric if it can be expressed in the form

(4)

where

is an Orthogonal Matrix and

is a Diagonal Matrix. This is equivalent to theMatrix equation

(5)

which is equivalent to

(6)

for all

, where

. Therefore, the diagonal elements of

are the Eigenvalues of

, and the columns of

are the corresponding Eigenvectors.

See also Antisymmetric Matrix, Skew Symmetric Matrix

References

Nash, J. C. ``Real Symmetric Matrices.'' Ch. 10 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 119-134, 1990.

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