In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors. Specifically the modal matrix M {\displaystyle M} for the matrix A {\displaystyle A} is the n × n matrix formed with the eigenvectors of A {\displaystyle A} as columns in M {\displaystyle M} . It is utilized in the similarity transformation
D = M − 1 A M , {\displaystyle D=M^{-1}AM,}
where D {\displaystyle D} is an n × n diagonal matrix with the eigenvalues of A {\displaystyle A} on the main diagonal of D {\displaystyle D} and zeros elsewhere. The matrix D {\displaystyle D} is called the spectral matrix for A {\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 M {\displaystyle M} .
Example The matrix
A = ( 3 2 0 2 0 0 1 0 2 ) {\displaystyle A={\begin{pmatrix}3&2&0\\2&0&0\\1&0&2\end{pmatrix}}}
has eigenvalues and corresponding eigenvectors
λ 1 = − 1 , b 1 = ( − 3 , 6 , 1 ) , {\displaystyle \lambda _{1}=-1,\quad \,\mathbf {b} _{1}=\left(-3,6,1\right),}
λ 2 = 2 , b 2 = ( 0 , 0 , 1 ) , {\displaystyle \lambda _{2}=2,\qquad \mathbf {b} _{2}=\left(0,0,1\right),}
λ 3 = 4 , b 3 = ( 2 , 1 , 1 ) . {\displaystyle \lambda _{3}=4,\qquad \mathbf {b} _{3}=\left(2,1,1\right).}
A diagonal matrix D {\displaystyle D} , similar to A {\displaystyle A} is
D = ( − 1 0 0 0 2 0 0 0 4 ) . {\displaystyle D={\begin{pmatrix}-1&0&0\\0&2&0\\0&0&4\end{pmatrix}}.}
One possible choice for an invertible matrix M {\displaystyle M} such that D = M − 1 A M , {\displaystyle D=M^{-1}AM,} is
M = ( − 3 0 2 6 0 1 1 1 1 ) . {\displaystyle M={\begin{pmatrix}-3&0&2\\6&0&1\\1&1&1\end{pmatrix}}.}
Note that since eigenvectors themselves are not unique, and since the columns of both M {\displaystyle M} and D {\displaystyle D} may be interchanged, it follows that both M {\displaystyle M} and D {\displaystyle D} are not unique.
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