Question

Prove: If A is an n × n symmetric matrix all of whose
eigenvalues are nonnegative, then x^{T}Ax ≥ 0 for all
nonzero x in the vector space Rn.

Answer #1

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.

Suppose A is a 2x2 matrix whose eigenvalues all have negative
real parts. For the system x 0 = Ax, is the origin stable or
unstable?

For these two problems, use the definition of eigenvalues.
(a) An n × n matrix is said to be nilpotent if Ak = O
for some positive integer k. Show that all eigenvalues of a
nilpotent matrix are 0.
(b) An n × n matrix is said to be idempotent if A2 =
A. Show that all eigenvalues of a idempotent matrix are 0, or
1.

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.

3.35 Show that if A is an m x m symmetric matrix with its
eigenvalues equal to its diagonal elements, then A must be a
diagonal matrix.

Problem 3.2
Let H ∈ Rn×n be symmetric and idempotent, hence a projection
matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number.
Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and ﬁnd
the joint distribution of (u,v). 1 (c) Someone claims that u and v
are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ.
(e) Assume that 1 ∈ Im(H) and ﬁnd...

7) Let B be a matrix with a repeated zero eigenvalues. Then show
that B2 = 0 (the 2 × 2 zero matrix). Use this to show: if A has a
repeated eigenvalue λ0, then (A − λ0I) 2 = 0. (Hint: Use the fact
that Bv = 0 for some nonzero vector v)

A matrix A is symmetric if AT = A and skew-symmetric
if AT = -A. Let Wsym be the set of all symmetric
matrices and let Wskew be the set of all skew-symmetric
matrices
(a) Prove that Wsym is a subspace of Fn×n . Give a
basis for Wsym and determine its dimension.
(b) Prove that Wskew is a subspace of Fn×n . Give a
basis for Wskew and determine its dimension.
(c) Prove that F n×n = Wsym ⊕Wskew....

matrix A (nxn). Prove that the sum of the eigenvalues of a
matrix A equals to the sum of its diagonal elements (Aii) using the
similarity of transformation's notation.

5. (a) Prove that det(AAT ) = (det(A))2.
(b) Suppose that A is an n×n matrix such that AT = −A. (Such an A
is called a skew- symmetric matrix.) If n is odd, prove that det(A)
= 0.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 11 minutes ago

asked 14 minutes ago

asked 20 minutes ago

asked 20 minutes ago

asked 23 minutes ago

asked 31 minutes ago

asked 42 minutes ago

asked 56 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago