Question

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 that R(A + B) = R(A)? If so, prove it;
otherwise, give a counterexample

Answer #1

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...

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.

(7) Prove the following statements.
(c) If A is invertible and similar to B, then B is invertible
and A−1 is similar to B−1 .
(d) The trace of a square matrix is the sum of the diagonal
entries in A and is denoted by tr A. It can be verified that tr(F
G)=tr(GF) for any two n × n matrices F and G. Prove that if A and B
are similar, then tr A = tr B

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

Q.Let A and B be n × n matrices such that A = A^2,
B = B^2, and AB = BA = 0.
(a) Prove that rank(A + B) = rank(A) + rank(B).
(b) Prove that rank(A) + rank(In − A) = n.

If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A
and B are 2x2 matrices, can I use that to show that Det(A)Det(B) =
Det(AB) for any n x n matrix? If so how?

Let A, B ? Mn×n be invertible matrices. Prove the following
statement: Matrix A is similar to B if and only if there exist
matrices X, Y ? Mn×n so that A = XY and B = Y X.

The trace of a square n×nn×n matrix A=(aij)A=(aij) is the sum
a11+a22+⋯+anna11+a22+⋯+ann of the entries on its main
diagonal.
Let VV be the vector space of all 2×22×2 matrices with real
entries. Let HH be the set of all 2×22×2 matrices with real entries
that have trace 11. Is HH a subspace of the vector space
VV?
Does HH contain the zero vector of VV?
choose H contains the zero vector of V H does not contain the zero
vector...

Select all statements below which are true for all invertible
n×n matrices A and B
A. AB=BA
B. (A+B)^2=A^2+B^2+2AB
C. (In−A)(In+A)=In−A^2
D. 7A is invertible
E. (AB)^−1=A^−1*B^−1
F. A+A^−1 is invertible

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

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