# can two different matrices have same eigenvalues and eigenvectors

Let $A$ and $B$ $\in M^{n\times n}$ be matrices with $n$ distinct eigenvalues $\{\lambda_1,\ldots ,\lambda_n\}$ and corresponding eigenvectors $\{v_1,v_2,\ldots,v_n\}$. Is it necessary that $A$ and $B$ are equal?

Yes. Since there are $n$ distinct eigenvalues the corresponding eigenvectors form a basis for $\mathbb R^n$. So knowing the eigenvectors and eigenvalues completely determines what the linear transformation corresponding to the matrix is.

For a 2x2 matrix you can visualize the answer Henning Makholm using the eigencircle of the matrix:

If you scroll up from the bookmark Eigencircles of special transformations,

You will see that if both eigenvalues and eigenvectors are fixed the transformation is fully defined.

The two green vectors$$v_{A1}$$ and $$v_{A2}$$ are the eigenvectors of A in the inserted figure.

you can also go straight the articles on eigencircles:

Englefield, M. J., & Farr, G. E. (2006). Eigencircles of 2 x 2 Matrices. Mathematics Magazine Vol. 79 Oct.,2006, 281-289.