The relationship between eigenvalues and determinant and trace of matrices __ Mathematics knowledge

relationship between eigenvalues of matrices and determinant and trace

From:http://www.cnblogs.com/andyjee/p/3737592.html
The product of eigenvalues of matrices equals the determinant of matrices The sum of the eigenvalues of the matrices equals the traces of the matrices

The simple understanding proves as follows: the Vedic theorem of 1, two quadratic equations:

Think about this: x^2+bx+c=0 the sum of all the roots of this equation is equal to how much, the product of all the roots equals

2, the two-time equation to extend to N times:

For a unary n-th equation, its root is recorded as

So then you can think like this: (X-X1) (x-x2) (x-x3) ... (x-n_n) =0 The coefficients of all the roots of the equation and corresponding to the number of items on the left of the equation, the product of all the roots corresponding to the coefficients of the number of items after the equation was expanded.

Description

A one-dollar five-time equation is known:

According to the algebraic principle of Gauss: the upper form must be decomposed in the plural range, and x1, x2, X3, X4, and X5 are the roots of the polynomial in the plural range.

3, consider the eigenvalue problem of the matrix to set a n-order square matrix, consider the feature polynomial | The n-1 of a-λi|, the eigenvalue equation of matrix A: Det (a-λi) =0 (determinant expansion is not explained here, can refer to relevant data), we can find that, in addition to the main diagonal element product (Λ-A11) (λ-a22) ... (Λ-ann), the number of other expanded items is less than n-1. So the coefficient of n-1 is (Λ-A11) (λ-a22) ... (Λ-ann) The coefficient of λ^ (n-1), i.e.-(A11+a22+...+ann).
Eigenvalue is the root of a feature polynomial, and = A11+a22+...+ann is known by the Vedic theorem (the relationship between root and coefficient).

4. Reference documents:

http://www.zhihu.com/question/20533117

Http://baike.baidu.com/view/1166.htm?fr=aladdin