In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.

Specifically the modal matrix ${\displaystyle M}$ for the matrix ${\displaystyle A}$ is the n × n matrix formed with the eigenvectors of ${\displaystyle A}$ as columns in ${\displaystyle M}$. It is utilized in the similarity transformation

where ${\displaystyle D}$ is an n × n diagonal matrix with the eigenvalues of ${\displaystyle A}$ on the main diagonal of ${\displaystyle D}$ and zeros elsewhere. The matrix ${\displaystyle D}$ is called the spectral matrix for ${\displaystyle A}$. The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in ${\displaystyle M}$.

The matrix

has eigenvalues and corresponding eigenvectors

A diagonal matrix ${\displaystyle D}$, similar to ${\displaystyle A}$ is

One possible choice for an invertible matrix ${\displaystyle M}$ such that ${\displaystyle D=M^{-1}AM,}$ is

Note that since eigenvectors themselves are not unique, and since the columns of both ${\displaystyle M}$ and ${\displaystyle D}$ may be interchanged, it follows that both ${\displaystyle M}$ and ${\displaystyle D}$ are not unique.