Expressing a symmetric matrix in terms on eigenvalues/vectors

[han35]

Homework Statement

Generate a random 10 x 10 symmetric matrix A (already done in MATLAB) . Express A in the form

Homework Equations

## A = ## [itex]\displaystyle \sum_{j=1}^{10}λ_j(A)v_jv^T_j\ [/itex]

for some real vectors ##v_j, j = 1, 2, . . . , 10.##

The Attempt at a Solution

I'm pretty sure the solution has something to do with the eigenvectors of a symmetric matrix being orthogonal.

The whole sum is basically ##A = λ_1v_1v^T_1 + λ_2v_2v^T_2 . . . λ_{10}v_{10}v^T_{10}.##

If ##v_j## are an eigenvectors then we can express that as:

##A = Av_1v^T_1 + Av_2v^T_2 . . . Av_{10}v^T_{10} ##
##→ A = A( v_1v^T_1 + v_2v^T_2 . . . v_{10}v^T_{10} ) ##

But that means we need ## v_1v^T_1 + v_2v^T_2 . . . v_{10}v^T_{10} = I ##. But I don't know if these vectors are meant to have those properties. =S

Basically I need to know what the ##v_j ##'s are

 

[han35]

Ok, so after testing some symmetric 3x3 matrices and computing it by hand I can confirm that the ##v_j##'s are indeed the eigenvectors of ##A##.

Now I just need to know why, lol.

 

[han35]

Does anyone know why?

If A is a symmetric matrix then A can be expressed as:

## A = ## [itex]\displaystyle \sum_{j=1}^{n}λ_j(A)v_jv^T_j\ [/itex]

where ##v_j, j = 1, 2, . . . , n.## are the eigenvectors of ##A##

But why?

 

[Dick]

Does anyone know why?

If A is a symmetric matrix then A can be expressed as:

## A = ## [itex]\displaystyle \sum_{j=1}^{n}λ_j(A)v_jv^T_j\ [/itex]

where ##v_j, j = 1, 2, . . . , n.## are the eigenvectors of ##A##

But why?

Your ##v_j## are an orthonormal set of eigenvectors, right? Multiply ##sum_{j=1}^{n}λ_j(A)v_jv^T_j## by any eigenvector ##v_k##. You get the same thing as multiplying that vector by A, right? If you want a fancy name for it, it's a simple version of the spectral theorem.