Any linear algebra genius in here? I will pay 1k fg to whoever can help me solve a problem.


12.
(a) State the Spectral Theorem for symmetric matrices. (Theorem 8.26)

(b) If A is symmetric prove that eigenvectors associated to distinct eigenvalues of A are orthogonal.

(c) If
A =
1 2 <---------- this part is a matrix
2 4  <----------
find the spectral factorization of A.

(d) Find the spectral decomposition of A.

This post was edited by mrbabydaddy on Dec 18 2016 11:48pm

(a) Maybe this one ?

For every symmetric real matrix A there exists a real orthogonal matrix Q such that D = (QT)AQ is a diagonal matrix,

where (QT) denotes the transpose of matrix Q.

(b) If A is symmetric, then <AX,Y> = <X,AY> for all X,Y in R^n.

Suppose now that X is an eigenvector associated to eigenvalue λ, and Y is an eigenvector associated to eigenvalue μ (μ≠λ).
Then λ.<X,Y> = <λX,Y> = <AX,Y> = <X,AY> = <X,μY> = μ.<X,Y>

If <X,Y> ≠ 0, then λ = μ, which is false.
Hence : <X,Y> = 0 (X, Y are orthogonal).

(c) Not sure if this is what you're calling "spectral factorization" :

| 1-x 2 |
| 2 4-x | = (1-x)(4-x) - 4 = x² - 5x = x(x-5)

hence λ = 0 and μ = 5 are the two eigenvalues of A.

(d)
Solve AX = 0 : X = (-2t , t ), then choose the eigenvector ( -2 , 1 ).
Solve AY = 5Y : Y = (t,2t), then choose the eigenvector ( 1 , 2 ).

Let Q = ( X Y ) = matrix ( -2 1 // 1 2 )

Q =
( -2 1 )
( 1 2 )

Compute Q⁻¹ = (1/5) * Q

then check that Q⁻¹ A Q = Diag(0,5).