Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only...

Question

Question

Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only possible eigenvalues of A are 0 and 1. (b) Prov

math Advanced-Math

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

# Let A be symmenic idampotent mabix. __je. Az =A! Now, suppose , is an eigen value of A. Then there is an eigen vector oc su Also, the multiplicity of 1 as an eigen value is precisely the ranks Hence rank(A) =). je brace (A) = ran(A). Hence proved. -

Please ask if you have doubts.

It's kind request to you to please upvote as it helps me alot.

Have a nice day

Add a comment

Answer 2

Similar Homework Help Questions

5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let...

5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let A be an arbitrary 2x2 matrix. Show that the matrix AA' is symmetric. Again, to prove these results you cannot use specific examples.)
5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let...

5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let A be an arbitrary 2x2 matrix. Show that the matrix AA' is symmetric (Again, to prove these results you cannot use specific examples.) 6) Let B I-A(A'A) A. a) Must B be square? Must A be square? Must (A'A) be square? b) Show that matrix B is idempotent. (Once again, do not use specific examples.)
Let A be an invertible matrix, prove that A is symmetric if and only if A-1...

Let A be an invertible matrix, prove that A is symmetric if and only if A-1 is symmetric.

Show each of the 3 following matrices is symmetric and idempotent ( J is a matrix...

Show each of the 3 following matrices is symmetric and idempotent ( J is a matrix with all 1 s) For the next few problems, let X = (X1X2), ßT = (β.β;), H the lat matrix for X, and Hi the hat matrix for X i. (I 1/nJ) ii. (I H)
2. Let A be an n x n real symmetric matrix or a complex normal matrix....

2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
(I) A square matrix E E M,xn(R) is idempotent if E-E. It is symmetric if E-E RR -[projyl& of projy relative to the standard basis (a) Let V C R be a subspace of R", and consider thé ortho...

(I) A square matrix E E M,xn(R) is idempotent if E-E. It is symmetric if E-E RR -[projyl& of projy relative to the standard basis (a) Let V C R be a subspace of R", and consider thé orthogonal projection projy onto V. Show that the representing matrix E & of IRn is both idempotent and symmetric. (b) Let E E Mnxn(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace VCR" such that...

A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matr...

A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matrix E = [projV]EE of proj V relative to the standard basis E of Rn is both idempotent and symmetric. (b) Let E∈Mn×n(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace V⊆Rn such that E= [projV]EE. [Hint: What...
Please solve both parts of this question! I've stared at it for a long time without knowing how to approach it. (1) A square matrix E є м,xn(R) is idempotent if E-E. It is symmetric if -t E. (a)...

Please solve both parts of this question! I've stared at it for a long time without knowing how to approach it. (1) A square matrix E є м,xn(R) is idempotent if E-E. It is symmetric if -t E. (a) Let V C Rn be a subspace of R, and consider the orthogonal projection projy R" Rn onto V. Show that the representing matrix E = projy18 of proj v relative to the standard basis of IRn is both idempotent and...
6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the...

6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the properties A",4 and A#AA. Use these properties to show that OS a“ 1, where a" is the ith diagonal element of A. [7 points] 6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the properties A",4 and A#AA. Use these properties to show that OS a“ 1, where a" is the ith...

Q22. Let A be an n x n symmetric matrix (so AT-A). Let a and b...

Q22. Let A be an n x n symmetric matrix (so AT-A). Let a and b be different eigenvalues of A, and let u and be eigenvectors for a and b, so Au au and 2y 2) Prove that u and g are orthogonal to each other. Hint. (Start with the expres- sion (Au,), and try simplifying it in a couple of different ways.)

Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only...

Homework Answers

Add Answer to:
Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only...

Post as a guest

Earn Coins

5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let...

5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let...

Let A be an invertible matrix, prove that A is symmetric if and only if A-1...

Show each of the 3 following matrices is symmetric and idempotent ( J is a matrix...

2. Let A be an n x n real symmetric matrix or a complex normal matrix....

(I) A square matrix E E M,xn(R) is idempotent if E-E. It is symmetric if E-E RR -[projyl& of projy relative to the standard basis (a) Let V C R be a subspace of R", and consider thé ortho...

A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matr...

Please solve both parts of this question! I've stared at it for a long time without knowing how to approach it. (1) A square matrix E є м,xn(R) is idempotent if E-E. It is symmetric if -t E. (a)...

6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the...

Q22. Let A be an n x n symmetric matrix (so AT-A). Let a and b...

Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only...

Homework Answers

Add Answer to: Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only...

Post as a guest

Earn Coins

Add Answer to:
Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only...