Question

**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.

Answer #1

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).

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.

1. If A is an n n matrix, prove that
(a) ATA is a symmetric matrix.
(b) A + AT is a symmetric matrix and A -
AT is a skew-symmetric matrix.
(c) A is the sum of a symmetric and a skew-symmetric matrix.

For the general case of n masses coupled by n
+1 springs, the mass matrix M
1)
is a (n+1) x (n+1) diagonal matrix (i.e. all off-diagonal
elements are identically equal to zero).
2)
is a (n+1) x (n+1) non-diagonal matrix (i.e it has non-zero
off-diagonal elements).
3)
is a (n) x (n) diagonal matrix (i.e. all off-diagonal elements
are identically equal to zero).
4)
is a (n) x (n) non-diagonal matrix (i.e it has non-zero
off-diagonal elements).

Recall that if A is an m × n matrix and B is a p × q matrix,
then the product C = AB is defined if and only if n = p, in which
case C is an m × q matrix. (a) Write a function M-file that takes
as input two matrices A and B, and as output produces the product
by columns of the two matrix. For instance, if A is 3 × 4 and B is...

Linear Algebra: Show that the set of all 2 x 2 diagonal matrices
is a subspace of M 2x2.
I know that a diagonal matrix is a square of n x n matrix whose
nondiagonal entries are zero, such as the n x n identity
matrix.
But could you explain every step of how to prove that this
diagonal matrix is a subspace of M 2x2.
Thanks.

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...

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...

Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m
R^n is a linear transformation

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...

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