Question

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.

Answer #1

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...

Let A, B be n × n matrices. The following are two incorrect
proofs that ABhas the same non-zero eigenvalues as BA. For each,
state two things wrong with the proof: (i) We will prove that AB
and BA have the same characteristic equation. We have that
det(AB − λI) = det(ABAA−1 − λAA−1) = det(A(BA − λI)A−1)
= det(A) + det(BA − λI) − det(A)
= det(BA − λI)
Hence det(AB − λI) = det(BA − λI), and so...

I'm studying Linear Algebra.
A Matrix A is a 3*3 matrix and it has 3 linear independent
eigenvectors. The Matrix B has the same eigenvectors (but not
necessarily the same eigenvalues). I'm supposed to prove that
AB=BA. I'm given a clue that says: ,, Is it possible to diagonal A
and B?''. I want to be clear that the matrixes are not given.
Thank you.

Let A be a (n × n) matrix. Show that A and AT have
the same characteristic polynomials (and therefore the same
eigenvalues). Hint: For any (n×n) matrix B, we have
det(BT) = det(B). Remark: Note that, however, it is
generally not the case that A and AT have the same
eigenvectors!

n×n-matrix M is symmetric if M = M^t. Matrix M is
anti-symmetric if M^t = -M.
1. Show that the diagonal of an anti-symmetric matrix are
zero
2. suppose that A,B are symmetric n × n-matrices. Prove that AB
is symmetric if AB = BA.
3. Let A be any n×n-matrix. Prove that A+A^t is symmetric and A
- A^t antisymmetric.
4. Prove that every n × n-matrix can be written as the sum of a
symmetric and anti-symmetric matrix.

For an n×n matrix, A, the trace of A is defined as the sum of
the entries on the main diagonal. That is, tr(A)=a11+a22+?+ann.
(a) Prove that for any matrices A and B having the same size,
tr(A+B)=tr(A)+tr(B) and for any scalar c, tr(cA)=ctr(A)
(b) Prove tr(A)=tr(AT) for all square matrices A.
(c) Prove that for any matrices A and B having the same size,
tr(AB)=tr(BA).
(d) Using (c), prove that if A and B are similar
tr(A)=tr(B).

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...

Suppose A and B are invertible matrices in Mn(R) and
that A + B is also invertible. Show that C = A-1 +
B-1 is also invertible.

We say the a matrix A is similar to a matrix B if there is some
invertible matrix P so that B=P^-1 AP.
Show that if A and B are similar matrices and b is an eigenvalue
for B, then b is also an eigenvalue for A. How would an eigenvector
for B associated with b compare to an eigenvector for A?

Show that the set GLm,n(R) of all mxn matrices with
the usual matrix addition and scalar multiplication is a finite
dimensional vector space
with dim GLm,n(R) = mn.
Show that if V and W be finite dimensional vector spaces with
dim V = m and dim W = n, B a basis for V and C a basis for W
then
hom(V,W)-----MatB--->C(-)-------->
GLm,n(R) is a bijective linear transformation. Hence or
otherwise, obtain dim hom(V,W).
Thank you!

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