Are the necessary and sufficient conditions for matrix equivalence and matrix similarity the same rank?

If a and B are both real symmetric matrices, then a and B are similar if and only if a and B have the same eigenvalues. In addition, if there is an invertible matrix P such that (...)

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1. What is the meaning and definition of equivalence of two matrices? What is the relationship between equivalence and similarity of two matrices? What are the necessary and sufficient conditions for equivalence of two matrices? What are the properties of equivalence?
2. 1. How to judge whether a matrix is similar to a diagonal matrix? 2. What kind of matrix is diagonal?
3. A necessary and sufficient condition for diagonal matrix Excuse me? Why is it necessary and sufficient for a diagonal matrix to be both an upper triangular matrix and a lower triangular matrix?
4. It is known that a and B are matrices of order n, and E-Ab is invertible. It is proved that e-Ba is invertible Counter proof: if e-Ba is irreversible, (e-Ba) x = 0, and the equation has non-zero solution, how to show that (E-Ab) x = 0 also has non-zero solution, and then the determinant of E-Ab is 0, which shows that E-Ab is irreversible and contradicts the known conditions, so the original proposition holds. How to show that (E-Ab) x = 0 also has non-zero solution?
5. Let a, B, a + B be invertible matrices of order n, prove that a ^ - 1 + B ^ - 1 is invertible, and find the inverse of a ^ - 1 + B ^ - 1,
6. Let a and B be invertible matrices of order n, and prove that (a *) * = | a | ^ n-2 · a
7. It is proved that there is an invertible matrix in a B, if a is invertible, then R (AB) = R (b) = R (BA)
8. Proof of rank of linear algebraic matrix and adjoint matrix At (3), I really don't understand why when R (a) & lt; n-1, all the N-1 order subformulas of a are 0? Why (3) step, R (a)
9. A linear algebra: A is a matrix of order n, R (a) = R
10. In linear algebra, if the rank of a matrix is increased by 1 or unchanged, how to prove it
11. If a and B are real symmetric matrices, it is necessary and sufficient that a and B have the same eigenvalues. Why?
12. The necessary and sufficient condition for the equivalence of matrix A and B is rank equality
13. Let a be a full rank matrix, B and C be n * t matrices. It is proved that the sufficient and necessary condition for ab = BC is b = C
14. What are the necessary and sufficient conditions for a matrix to be similar to B? AB is an arbitrary matrix. It is not specified that AB is a real symmetric matrix or diagonalizable. If necessary, it can be regarded as a part of the necessary and sufficient conditions
15. How to find the last step of diagonalization of similar matrix?
16. A problem of diagonalization of matrix A=|1 -1 1 | |2 4 -2| |-1 2 0|
17. If two matrices are similar, can they be diagonalized? In other words, can diagonalize the matrix and its similar matrix? Best examples
18. Let the eigenvalues of a matrix of order 3 be 1,2, - 2, and B = 3a2-a3, then find the diagonal matrix with eigenvalues similar to B | B | a-3i |? (the number after a is superscript)
19. A square matrix of order n has n different eigenvalues, which is similar to a diagonal matrix () A. Sufficient and necessary condition B. sufficient but not necessary condition C. necessary but not sufficient condition D. neither sufficient nor necessary condition
20. The sufficient condition for the similarity of n-order matrix A and diagonal matrix is that a has n different eigenvalues and a is a real symmetric matrix. I want to ask: the general problem is to prove that n-order matrix A and B are similar. In this way, is it necessary to prove matrix B first It can be diagonalized, and then the above sufficient conditions are used to prove the similarity