Question

Let H = X(X'X)^-1X' and M = In - X(X'X)^-1X' where X is a n x...

Let H = X(X'X)^-1X' and M = In - X(X'X)^-1X' where X is a n x k-dimensional matrix.

Show that H is positive semi-definite and has rank k. Use that a symmetric matrix is positive semi-definite if and only if its eigenvalues are non-negative. You can use that rank(H) = trace(H).

Math   Advanced-Math   
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 an m × n matrix, and Q be an n × n invertible...
Let A be an m × n matrix, and Q be an n × n invertible matrix. (1) Show that R(A) = R(AQ), and use this result to show that rank(AQ) = rank(A); (2) Show that rank(AQ) = rank(A).
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...
3.35 Show that if A is an m x m symmetric matrix with its eigenvalues equal...
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.
Let N and H be groups, and here for a homomorphism f: H --> Aut(N) =...
Let N and H be groups, and here for a homomorphism f: H --> Aut(N) = group automorphism, let N x_f H be the corresponding semi-direct product. Let g be in Aut(N), and  k  be in Aut(H),  Let C_g: Aut(N) --> Aut(N) be given by conjugation by g.  Now let  z :=  C_g * f * k: H --> Aut(N), where * means composition. Show that there is an isomorphism from Nx_f H to Nx_z H, which takes the natural...
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.
Let A be an n × n matrix and let x be an eigenvector of A...
Let A be an n × n matrix and let x be an eigenvector of A corresponding to the eigenvalue λ . Show that for any positive integer m, x is an eigenvector of Am corresponding to the eigenvalue λ m .
Prove the following: Given k x m matrix A, m x n matrix B. Then rank(A)=m...
Prove the following: Given k x m matrix A, m x n matrix B. Then rank(A)=m --> rank(AB)=rank(B)
Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m...
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
In the system AX=b, where A is m x n matrix and rank of A is...
In the system AX=b, where A is m x n matrix and rank of A is m, you are given n vectors and among them p vectors are linearly dependent (p > m). Please write down the procedure to reduce the number of dependent vector by 1.
Prove that the quadratic form q(h) = hMh is indefinite if and only if M has...
Prove that the quadratic form q(h) = hMh is indefinite if and only if M has both positive and negative eigenvalues.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT