Mathematical problem related to eigen values and eigen vectors

[Roshdy]

Dear all
it is well known that
A=∑ λ e eT
such that λ, e, eT are the eigen values, eigen vectors and eigen vector transpose of A respectively

The problem
if given that A=x.xT, such that x is a column vector and xT is its transpose.

can x be represented in terms of eigen values and eigen vectors λ and e

 

[LouisSheffield]

Mathematical problem

Roshdy -

I "tried" it by filling vectors with non-complex noise.
This is far from conclusive, but it might shorten your attempts to find the "real" answer.

In short, yes ... but the largest eigenvalue was always the same as the scalar product (i.e. the only non-zero eigenvalue of the outer product is the inner product).

As such, Sqrt(lamda) x the eigenvector recovers your original vector that you formed outer and inner products with.

Hope this helps at least some ...

 

[LouisSheffield]

Re: Mathematical problem

Roshdy - I had that one a little bit off - at least "mirky" - I'll try again:

For any vector x, xT.x is the inner product, and is also a valid (possibly the ONLY) eigenvalue. So, let's assume that λ = xT.x

Now, the eigenvector associated with this λ is equal to e = 1/√λ . x
Equally, √λ . e = x

If √λ . e = x, then we must also have (λ is just a scaling factor) √λ . eT = xT

Each of the vectors, e and eT, are "off from x and xT" by the factor √λ which,
being the inner product, is already in a "squared nature".
(√λ . e).(√λ . eT) recovers x.xT