Recovering original matrix from Eigenvalue Decomposition

Question

According to Wikipedia the eigenvalue decomposition should be such that: http://en.wikipedia.org/wiki/Square_root_of_a_matrix See section Computational Methods by diagonalization: Sp that if matrix A is decomposed such that it has Eigenvector V and Eigenvalues D, then A=VDV'. ``` A=[1 2; 3 4]; [V,D]=eig(A); RepA=V*D*V'; ``` However in Matlab, A and RepA are not equal? Why is this? Baz

Accepted Answer

In general, the formula is:

```
RepA = V*D*inv(V);
```

or, written for better numeric accuracy in MATLAB,

```
RepA = V*D/V;
```

When A is symmetric, then the V matrix will turn out to be orthogonal, which will make `inv(V) = V.'`. A is NOT symmetric, so you need the actual inverse.

Try it:

```
A=[1 2; 2 3];  % Symmetric
[V,D]=eig(A);
RepA = V*D*V';
```

Recovering original matrix from Eigenvalue Decomposition

Solution

Problem