Question

Let A be an n × n-matrix. Show that there exist B, C such that B...

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

Math   Algebra   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let A be a (n × n) matrix. Show that A and AT have the same...
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...
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...
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...
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....
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.
Problem 3.2 Let H ∈ Rn×n be symmetric and idempotent, hence a projection matrix. Let 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...
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...
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,...
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.
4. Suppose that we have a linear system given in matrix form as Ax = b,...
4. Suppose that we have a linear system given in matrix form as Ax = b, where A is an m×n matrix, b is an m×1 column vector, and x is an n×1 column vector. Suppose also that the n × 1 vector u is a solution to this linear system. Answer parts a. and b. below. a. Suppose that the n × 1 vector h is a solution to the homogeneous linear system Ax=0. Showthenthatthevectory=u+hisasolutiontoAx=b. b. Now, suppose that...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT