# how to find find a matrix by the characteristic vlaues and vectors

Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen values) and characteristic vectors (eigen vectors). But my question is why cant we find a matrix by the characteristic values and vectors ?

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Is this a question about mathematics, or the software Mathematica? This site is for the latter. –  Simon Woods Apr 16 at 14:14

You should note that for a diagonalizable matrix $\mathbf{M}$, the following equality holds:
$$\mathbf{M}=\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{-1}$$
Where $\mathbf{\Lambda}=\operatorname{diag}(\lambda_{1},\cdots,\lambda_{n})$, and $\lambda_{i}$ is the $i$th eigenvalue; and $\mathbf{P}$ is the matrix formed by the eigenvectors, i.e:
$$\mathbf{P}=\begin{pmatrix}\uparrow && \uparrow \\ v_{1} & \cdots & v_{n} \\ \downarrow && \downarrow\end{pmatrix}$$
You can use this to find $\mathbf{M}$ from the eigenvalues and eigenvectors.