Question

Problem 30. Show that if two matrices A and B of the same size have the property that Ab = Bb for every column vector b of the correct size for multiplication, then A = B.

Answer #1

Assume A and B are two nonsingular square matrices.
Prove that AB has the same eigenvalue as BA.

Let matrices A,B∈Mn×n(R). Show that if A and B are each similar
to some diagonal matrix, and also have the same eigenvectors (but
not necessarily the same eigenvalues), then AB=BA.

If A and B are matrices, A has size m × n, and the
multiplication ABA is defined, what can you say about the size of
B, if anything?

If A and B are matrices and the columns of AB are independent,
show that the columns of B are independent.

Show that the product of two n × n unitary matrices is unitary.
Is the same true of the sum of two n × n unitary matrices? Prove or
find a counterexample.

We say two n × n matrices A and B are similar if there is an
invertible n × n matrix P such that
A = PBP^ -1.
a) Show that if A and B are similar n × n matrices, then they
must have the same determinant.
b) Show that if A and B are similar n × n matrices, then they
must have the same eigenvalues.
c) Give an example to show that A and B can be...

If A and B are conjugate
matrices they have the same eigenvectors.

Prove the following statements:
a) If A and B are two positive semideﬁnite matrices in IR ^ n ×
n , then trace (AB) ≥ 0. If, in addition, trace (AB) = 0, then AB =
BA =0
b) Let A and B be two (diﬀerent) n × n real matrices such that
R(A) = R(B), where R(·) denotes the range of a matrix.
(1) Show that R(A + B) is a subspace of R(A).
(2) Is it always true...

A and B are two m*n matrices. a. Show that B is invertible. b.
Show that Nullsp(A)=Nullsp(BA)

Hi, I know that if two matrices A and B are similar matrices
then they must have the same eigenvalues with the same geometric
multiplicities. However, I was wondering if that statement was
equivalent. In order terms, if two matrices have the same
eigenvalues with the same geometric multiplicities, must they be
similar? If not, is it always false?

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