Question

A square matrix A is said to be idempotent if A2 = A. Let A be...

A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.

  1. Show that I − A is also idempotent.
  2. Show that if A is invertible, then A = I.
  3. Show that the only possible eigenvalues of A are 0 and 1.(Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)
  4. Let W = col(A). Show that TA(x) = projW x and TI−A(x) = projW ⊥ x.
Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

I hope you will get it

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
A square matrix A is said to be idempotent if A2 = A. Let A be...
A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix. Show that I − A is also idempotent. Show that if A is invertible, then A = I. Show that the only possible eigenvalues of A are 0 and 1.(Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.) Let W = col(A). Show that TA(x) = projW x and TI−A(x)...
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 .
Let P be a stochastic matrix. Show that λ=1 is an eigenvalue of P. What is...
Let P be a stochastic matrix. Show that λ=1 is an eigenvalue of P. What is the associated eigenvector?
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...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT