Let A ∈ Mat n×n(R) be a real square matrix.
(a) Suppose that A is symmetric, positive semi-definite, and
orthogonal. Prove that A is the identity matrix.
(b) Suppose that A satisfies A = −A^T . Prove that if λ ∈ C is
an eigenvalue of A, then λ¯ = −λ.
From now on, we assume that A is idempotent, i.e. A^2 = A.
(c) Prove that if λ is an eigenvalue of A, then λ is equal to 0...
Divide and Conquer (Strassen’s Matrix
Multiplication)
Given two square matrices A and B of size n x n each, find their
multiplication matrix.
Naive Method
Following is a simple way to multiply two matrices.
void multiply(int A[][N],
int B[][N], int C[][N]) {
for
(int i = 0; i < N; i++) {
for
(int j = 0; j < N; j++) {
C[i][j]
= 0;
for
(int k = 0; k < N; k++) {
C[i][j]
+= A[i][k]*B[k][j];
}...
Suppose C is a m × n matrix and A is a n × m matrix. Assume CA =
Im (Im is the m × m identity matrix). Consider the n × m system Ax
= b.
1. Show that if this system is consistent then the solution is
unique.
2. If C = [0 ?5 1
3 0 ?1]
and A = [2 ?3
1 ?2
6 10] ,,
find x (if it exists) when
(a) b =[1...
Linear Algebra
Carefully prove the following statement:
Let A be an n×n matrix. Assume that there exists an integer k ≥
1 such that Ak = I . Prove that A is invertible.
Use this theorem to find the inverse of the given matrix or show
that no inverse exists. (If an answer does not exist, enter DNE in
any cell.)
1
2
5
1
−1
0
2
1
2
1
−5
0
1
1
2
1
Let A be an m × n matrix and B be an m × p matrix. Let C =[A |
B] be an m×(n + p) matrix.
(a) Show that R(C) = R(A) + R(B), where R(·) denotes the range
of a matrix.
(b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).
A square matrix A is said to be symmetric if its transpose
AT satisfies AT= A, and a
complex-valued square matrix A is said to be Hermitian if its
conjugate transpose AH =
(A)T = AT satisfies AH = A. Thus,
a real-valued square matrix A is symmetric if and
only if it is Hermitian. Which of the following is a vector
space?
(a) The set of all n xn real-valued symmetric matrices over
R.
(b) The set of all...