# Math Help - Reconstruct a matrix from its reduced eigenparis

1. ## Reconstruct a matrix from its reduced eigenparis

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 put in a descending order.

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!!

2. ## Re: Reconstruct a matrix from its reduced eigenparis

I'm not sure what you mean by "reliably". Your reconstruction is going to be lossy.

That being said if you've decomposed your original matrix as

$A=Q \Lambda Q^{-1}$

where $Q$ is the matrix of eigenvector columns and

$\Lambda$ is the diagonal matrix of eigenvalues

you should be able to just zero out the eigenvalues below some fidelity threshold, thus forming $\Lambda_{reduced}$.

Then

$A_{reduced}=Q \Lambda_{reduced} Q^{-1}$