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Finding the rank of matrix (2 - 1 - 1 1 2) (1 1 - 2 1 4) (4 - 6 2) by matrix transformation- Finding the rank of matrix by matrix transformation （2 -1 -1 1 2） （1 1 -2 1 4） （4 -6 2 -2 4） （3 6 -9 7 9） How to find rank, some matrix transformation is hard to think of
1、 The matrix A is regarded as a column vector, and is written as a matrix composed of column vectors: 2,1,4,3, - 1,1, - 6,6, - 1, - 2,2, - 9,1,1, - 2,7,2,4,4,9. Second, exchange the first and fourth rows, and do not change the rank of the matrix: 1,1, - 2,7, - 1,1, - 6,6, - 1, - 2,2, - 9,2,1,4,3,2,4,9
Finding P in the transformation of congruent matrix
It is known that a is a real symmetric matrix and B is a diagonal matrix. A and B are congruent but not similar. Find the invertible matrix P so that p'ap = B. (p'is the transpose matrix of P). Want to know the general process of solving P
Constructing block matrix
A
E
At the same time, the upper half of the matrix is transformed into B by the elementary column transformation (and the corresponding elementary row transformation for the upper half)
Finally, it turns into
B
P
Then p is the result
Finding the transformation matrix P of Jordan canonical form
A=
[3,2,1
0,4,0
-1,2,5]
P^(-1)AP=J
There is no solution,
First, the eigenvalues of a are calculated to be 4,4,4, and then
A-4I=
-1 2 1
0 0 0
-1 2 1
So J should have a first-order block and a second-order block
Suppose P = [P1, P2, P3], J=
4 0 0
0 4 1
0 0 4
Then (a-4i) P = P (j-4i), we can know that P1 and P2 are eigenvectors, (a-4i) P3 = P2, so P2 should be taken from the image space of a-4i, for example, P2 = [1,0,1] ^ t, then P3 = [0,0,1] ^ t,
Finally, take another solution of (a-4i) x = 0, such as P1 = [2,1,0] ^ t, and we get the transformation matrix P
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