Find the eigenvalues and eigenvectors of the matrix A = the first row 2, - 1,1, the second row 0,3, - 1, the third row 2,1,3

Find the eigenvalues and eigenvectors of the matrix A = the first row 2, - 1,1, the second row 0,3, - 1, the third row 2,1,3

A = [2,-1,1;0,3,-1;2,1,3];
>> [V,S] = eig(A)
V =
-0.5774 -0.5774 + 0.0000i -0.5774 - 0.0000i
0.5774 0.5774 0.5774
-0.5774 0.5774 - 0.0000i 0.5774 + 0.0000i
S =
4.0000 0 0
0 2.0000 + 0.0000i 0
0 0 2.0000 - 0.0000i

Find the eigenvalues and eigenvectors of the following matrix A, the first line 1 23, the second line 2 13, the third line 2 35

There are three eigenvalues
Eigenvalue 1:8
Eigenvalue 2:0
Eigenvalue 3: - 1
Eigenvectors:
Vector 1 vector 2 vector 3
0.4575 0.5774 -0.4286
0.4575 0.5774 0.8571
0.7625 -0.5774 -0.2857

Matrix A = first row 1,2,3, second row 2,1,3, third row 3,3,6. If there are two eigenvalues 0,9, find the third eigenvalue. (fill in the blank)

The sum of diagonal elements of matrix (trace of square matrix) is equal to the sum of eigenvalues
1 + 1 + 6 - 0 - 9 = -1
That's - 1

Given the matrix A, find the invertible matrix P so that PA is the simplest form of rows. Is p unique

The simplest form of a row is unique
When a is reversible, P is unique
When a is irreversible, P is not unique

If P is an invertible matrix, then R (PA) = R (a)
How to prove it
cnheying：
You mean matrix A becomes PA after a finite elementary transformation, right?
After elementary transformation, the rank of matrix remains unchanged.

If P is invertible, then p can be decomposed into the product of Elementary Matrices of finite degree
Let P = P1 * P2 * p3.pn
PA=P1(P2(...(PnA)))

What is the relationship between a and its row simplest matrix B? How to find the invertible matrix P so that PA = B?

The matrix A and an identity matrix with the same number of rows are put together, that is, (a, e). The elementary row transformation is applied to this matrix
When its row simplest matrix B, e is reduced to the required invertible matrix P such that PA = B

If the product of two matrices AB is a zero matrix and it is known that a is not a zero matrix, can we conclude that B is a zero matrix?
When three matrices are not zero matrices, why can we use the error theory mentioned above

No. the multiplication of matrix has zero factor and does not satisfy the elimination law
How can the above conclusions be used?

The condition of nonzero matrix product being zero

A necessary and sufficient condition for ab = 0
If all the column vectors in B are solutions of AX = 0

If the product of matrix A and B AB = 0, and a ≠ 0, then there must be B = 0, is it correct

It's not right
The column vector of B is the solution of AX = 0

Given that the product of two nonzero matrices is a zero matrix, it is proved that the two matrices are irreversible

AB=O
To the contrary:
If a is reversible, then (B is reversible)
Multiply both sides by a ^ (- 1)
A^(-1)AB=A^(-1)O
B=O
And the matrix is nonzero
These two matrices are not invertible