Question

Let A be an n × n-matrix. Show that there exist B, C such that B is symmetric, C is skew-symmetric, and A = B + C. (Recall: C is called skew-symmetric if C + C^T = 0.) Remark: Someone answered this question but I don't know if it's right so please don't copy his solution

Answer #1

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!

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.

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....

Let A be an nxn matrix. Show that if Rank(A) = n, then Ax = b
has a unique solution for any nx1 matrix b.

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.

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.

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...

Let A be an m×n matrix, x a vector in Rn, and b a vector in Rm.
Show that if x1 in Rn is a solution to Ax=b and x2 is a solution to
Ax=⃗0, then x1 +x2 is a solution to Ax=b.

1. Let a ∈ Z and b ∈ N. Then there exist q ∈ Z and r ∈ Z with 0
≤ r < b so that a = bq + r.
2. Let a ∈ Z and b ∈ N. If there exist q, q′ ∈ Z and r, r′ ∈ Z
with 0 ≤ r, r′ < b so that a = bq + r = bq′ + r ′ , then q ′ = q
and r...

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.

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