Reconstruct A from its reduced eigenparis

[jollage]

Hi all,

Suppose I have a matrix $A_{N\times N}$. I compute its eigenmodes

$A V = V \Lambda$.

$V, \Lambda$ are eigenvectors and eigenvalues of size $N\times N$. The eigenvalues are descending.

Now I cut off several eigenmodes (the ones having small value), it becomes

$A V_{N \times n} = V_{N \times n} \Lambda_{n \times n}$.

What I want is to reconstruct $A$ from $V_{N \times n}, \Lambda_{n \times n}$.

It seems that in Matlab, if I just modify the above equation

$A = V_{N \times n} \Lambda_{n \times n}V^{-1}_{N \times n}$,

it will not work.

So my question is: how can I reliably reconstruct the original matrix by using its reduced eigenmodes? Thanks!

 

[AlephZero]

I don't think you can do this for an arbitrary matrix $A$. It's not clear what you mean by $V_{N\times m}^{-1}$ for a non-square matrix $V_{N\times m}$.

You can do something similar with the singular value decomposition $A = U\Sigma V^T$ where $U$ and $V$ are unitary matrices.

You can also do what you want if $A$ is Hermitian, because $V^{-1} = V^T$.