On the problem of finding diagonal matrix in linear algebra After a diagonalizable matrix is substituted into the characteristic equation λ e-A, the obtained λ is assumed to be three, so the elements on the main diagonal line of the diagonal matrix are also three. How can we judge which is the first and which is the second order of the three elements in the diagonal matrix, The element arrangement of diagonal matrix is also 822. If I get that λ 1 and λ 2 are 2 and λ 3 is 8, then the final diagonal matrix is not 228, but 660. T_ Why is this?

On the problem of finding diagonal matrix in linear algebra After a diagonalizable matrix is substituted into the characteristic equation λ e-A, the obtained λ is assumed to be three, so the elements on the main diagonal line of the diagonal matrix are also three. How can we judge which is the first and which is the second order of the three elements in the diagonal matrix, The element arrangement of diagonal matrix is also 822. If I get that λ 1 and λ 2 are 2 and λ 3 is 8, then the final diagonal matrix is not 228, but 660. T_ Why is this?

Obviously, the order of permutation can be arbitrary, which depends on the order of eigenvectors
If AP1 = 8p1, ap2 = 2P2, AP3 = 2P3, take the matrix P = (P1, P2, P3), then (P inverse) AP = diag (8,2,2)
If you choose P = (P2, P3, P1), then (P inverse) AP = diag (2,2.8)
If you just need to know what kind of diagonal matrix this diagonalizable matrix is similar to, as long as the diagonal elements are 8, 2 and 2, regardless of the order

Finding diagonal matrix in linear algebra
Find out the eigenvector and get P
Is it necessary to slowly multiply the three matrices p ^ (- 1) AP to get the diagonal matrix B
Is there a simpler way?

That's what I always do=

What are nonzero solutions in linear algebra?

If there are n unknowns, the system of equations composed of n equations: a11x1 + a12x2 +... + a1nxn = 0, a21x1 + a22x2 +... + a2nxn = 0,... An1x1 + an2x2 +... + annxn = 0. Obviously, if x1, x2.xn are equal to 0, we can make this

How to understand zero solution, non-zero solution, infinite solution and unique solution in linear algebra

The zero solution is X1 = x2 = X3 = =xn=0
Non zero solution means that XM is not equal to 0
An infinite solution is an infinite number of solutions, such as x 1 + x 2 = 0
The only solution is that there is only one set of solutions