l can be 0, but x cannot be 0; l is finite, but the number of x is infinite

The null space of (A-II) is the eigenspace corresponding to l

Otherwise, Ax=0 has only one solution, that is, 0 solution

The eigenvalue of A squared is l squared®AAx=lAx=llx

The eigenvalue of A-1 is 1/l®A-1Ax=A-1lx=x, so A-1x=1/l x

The characteristic polynomial of A is the expression of the determinant of (A-lI)

det(A-lI)=0 is the characteristic equation of A

The necessary and sufficient condition for the number l to be the eigenvalue of n×n matrix A is that l is the root of the characteristic equation det(A-lI)=0

The multiplicity of l as a root of a characteristic equation is called the (algebraic multiplicity) of l

If A and B are n×n matrices, and if there is an invertible matrix P, which is PAP-1=B, or equivalently A=PBP-1, then A is said to be similar to B or B is similar to A, that is, AB is similar.

The transformation that turns A into P-1AP is called similarity transformation.

If a square matrix A is similar to a diagonal matrix, that is, there is an invertible matrix P and a diagonal matrix D, and A=PDP-1, then A is said to be diagonalizable

For A=PDP-1, D is a diagonal matrix «The column vectors of P are n linearly independent eigenvectors of A. At this time, the elements on the main diagonal of D are the eigenvectors of A corresponding to P. eigenvalues

The necessary and sufficient condition for A to be diagonal is that there are enough eigenvectors to form the basis of Rn. Such a basis is called an eigenvector basis (en eigenvector basis).

Non-triangular matrices cannot be calculated by row transformation, which will affect the eigenvalues!

It is not a necessary and sufficient condition. It is possible to diagonalize if there are <n distinct eigenvalues

a. For 1≤k≤p, the dimension of the feature space of lk is less than or equal to the algebraic multiplicity of lk

b. The necessary and sufficient condition for matrix A to be diagonal is that the sum of the dimensions of all different feature spaces is n

c. If A is diagonalizable and Bk is the basis of the feature space corresponding to lk, then the set of all vectors in the set B1,...,Bp is the feature vector basis of Rn (the total number is n and linearly independent)

definition

core

When V=W, C=B, M is called the matrix of T relative to B, or is referred to as the B-matrix of T, denoted as [T]B

The coordinates of x under base B after transformation represent [T(x)]B = [T]B [x]B

[T]B=[ [T(b1)]B ... [T(bn)]B ] in the same space, the basis remains unchanged

Theorem 8 (diagonal matrix representation) Let A=PDP-1, where D is an n×n diagonal matrix. If the basis B of Rn consists of the column vector of P, then D is the B-matrix of the linear transformation x|®Ax