# Eigenvalues and Eigenvectors uniquely define a matrix

1. Jul 1, 2009

### zeebo17

Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix $$M$$ from a matrix of its eigenvectors as columns $$P$$ and a diagonal matrix of the eigenvalues $$E$$ through $$M=P E P^{\dagger}$$?

2. Jul 1, 2009

### ice109

i'm pretty sure the answer is yes

3. Jul 1, 2009

### 0xDEADBEEF

It is a bit more complicated. What you say is true if you can diagonalize the matrix. But take a matrix with complex eigenvalues and you are quickly missing eigenvectors. look up algebraic vs. geometric multiplicity of eigenvectors

4. Jul 1, 2009

### g_edgar

Try it with 2 x 2 matrix [0, 1; 0, 0]

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