Modal matrix

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.^[1]

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

${\displaystyle D=M^{-1}AM,}${\displaystyle D=M^{-1}AM,}

where ${\displaystyle D}${\displaystyle D} is an n ×ばつ n diagonal matrix with the eigenvalues of ${\displaystyle A}${\displaystyle A} on the main diagonal of ${\displaystyle D}${\displaystyle D} and zeros elsewhere. The matrix ${\displaystyle D}${\displaystyle D} is called the spectral matrix for ${\displaystyle A}${\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 ${\displaystyle M}${\displaystyle M}.^[2]

The matrix

${\displaystyle A={\begin{pmatrix}3&2&0\2円&0&0\1円&0&2\end{pmatrix}}}${\displaystyle A={\begin{pmatrix}3&2&0\2円&0&0\1円&0&2\end{pmatrix}}}

has eigenvalues and corresponding eigenvectors

${\displaystyle \lambda _{1}=-1,\quad ,円\mathbf {b} _{1}=\left(-3,6,1\right),}${\displaystyle \lambda _{1}=-1,\quad ,円\mathbf {b} _{1}=\left(-3,6,1\right),}

${\displaystyle \lambda _{2}=2,\qquad \mathbf {b} _{2}=\left(0,0,1\right),}${\displaystyle \lambda _{2}=2,\qquad \mathbf {b} _{2}=\left(0,0,1\right),}

${\displaystyle \lambda _{3}=4,\qquad \mathbf {b} _{3}=\left(2,1,1\right).}${\displaystyle \lambda _{3}=4,\qquad \mathbf {b} _{3}=\left(2,1,1\right).}

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

${\displaystyle D={\begin{pmatrix}-1&0&0\0円&2&0\0円&0&4\end{pmatrix}}.}${\displaystyle D={\begin{pmatrix}-1&0&0\0円&2&0\0円&0&4\end{pmatrix}}.}

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

${\displaystyle M={\begin{pmatrix}-3&0&2\6円&0&1\1円&1&1\end{pmatrix}}.}${\displaystyle M={\begin{pmatrix}-3&0&2\6円&0&1\1円&1&1\end{pmatrix}}.}^[3]

Note that since eigenvectors themselves are not unique, and since the columns of both ${\displaystyle M}${\displaystyle M} and ${\displaystyle D}${\displaystyle D} may be interchanged, it follows that both ${\displaystyle M}${\displaystyle M} and ${\displaystyle D}${\displaystyle D} are not unique.^[4]

Let ${\displaystyle A}${\displaystyle A} be an n ×ばつ n matrix. A generalized modal matrix ${\displaystyle M}${\displaystyle M} for ${\displaystyle A}${\displaystyle A} is an n ×ばつ n matrix whose columns, considered as vectors, form a canonical basis for ${\displaystyle A}${\displaystyle A} and appear in ${\displaystyle M}${\displaystyle M} according to the following rules:

• All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of ${\displaystyle M}${\displaystyle M}.
• All vectors of one chain appear together in adjacent columns of ${\displaystyle M}${\displaystyle M}.
• Each chain appears in ${\displaystyle M}${\displaystyle M} in order of increasing rank (that is, the generalized eigenvector of rank 1 appears before the generalized eigenvector of rank 2 of the same chain, which appears before the generalized eigenvector of rank 3 of the same chain, etc.).^[5]

One can show that

where ${\displaystyle J}${\displaystyle J} is a matrix in Jordan normal form. By premultiplying by ${\displaystyle M^{-1}}${\displaystyle M^{-1}}, we obtain

Note that when computing these matrices, equation (1 ) is the easiest of the two equations to verify, since it does not require inverting a matrix.^[6]

This example illustrates a generalized modal matrix with four Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.^[7] The matrix

${\displaystyle A={\begin{pmatrix}-1&0&-1&1&1&3&0\0円&1&0&0&0&0&0\2円&1&2&-1&-1&-6&0\\-2&0&-1&2&1&3&0\0円&0&0&0&1&0&0\0円&0&0&0&0&1&0\\-1&-1&0&1&2&4&1\end{pmatrix}}}${\displaystyle A={\begin{pmatrix}-1&0&-1&1&1&3&0\0円&1&0&0&0&0&0\2円&1&2&-1&-1&-6&0\\-2&0&-1&2&1&3&0\0円&0&0&0&1&0&0\0円&0&0&0&0&1&0\\-1&-1&0&1&2&4&1\end{pmatrix}}}

has a single eigenvalue ${\displaystyle \lambda _{1}=1}${\displaystyle \lambda _{1}=1} with algebraic multiplicity ${\displaystyle \mu _{1}=7}${\displaystyle \mu _{1}=7}. A canonical basis for ${\displaystyle A}${\displaystyle A} will consist of one linearly independent generalized eigenvector of rank 3 (generalized eigenvector rank; see generalized eigenvector), two of rank 2 and four of rank 1; or equivalently, one chain of three vectors ${\displaystyle \left\{\mathbf {x} _{3},\mathbf {x} _{2},\mathbf {x} _{1}\right\}}${\displaystyle \left\{\mathbf {x} _{3},\mathbf {x} _{2},\mathbf {x} _{1}\right\}}, one chain of two vectors ${\displaystyle \left\{\mathbf {y} _{2},\mathbf {y} _{1}\right\}}${\displaystyle \left\{\mathbf {y} _{2},\mathbf {y} _{1}\right\}}, and two chains of one vector ${\displaystyle \left\{\mathbf {z} _{1}\right\}}${\displaystyle \left\{\mathbf {z} _{1}\right\}}, ${\displaystyle \left\{\mathbf {w} _{1}\right\}}${\displaystyle \left\{\mathbf {w} _{1}\right\}}.

An "almost diagonal" matrix ${\displaystyle J}${\displaystyle J} in Jordan normal form, similar to ${\displaystyle A}${\displaystyle A} is obtained as follows:

${\displaystyle M={\begin{pmatrix}\mathbf {z} _{1}&\mathbf {w} _{1}&\mathbf {x} _{1}&\mathbf {x} _{2}&\mathbf {x} _{3}&\mathbf {y} _{1}&\mathbf {y} _{2}\end{pmatrix}}={\begin{pmatrix}0&1&-1&0&0&-2&1\0円&3&0&0&1&0&0\\-1&1&1&1&0&2&0\\-2&0&-1&0&0&-2&0\1円&0&0&0&0&0&0\0円&1&0&0&0&0&0\0円&0&0&-1&0&-1&0\end{pmatrix}},}${\displaystyle M={\begin{pmatrix}\mathbf {z} _{1}&\mathbf {w} _{1}&\mathbf {x} _{1}&\mathbf {x} _{2}&\mathbf {x} _{3}&\mathbf {y} _{1}&\mathbf {y} _{2}\end{pmatrix}}={\begin{pmatrix}0&1&-1&0&0&-2&1\0円&3&0&0&1&0&0\\-1&1&1&1&0&2&0\\-2&0&-1&0&0&-2&0\1円&0&0&0&0&0&0\0円&1&0&0&0&0&0\0円&0&0&-1&0&-1&0\end{pmatrix}},}

${\displaystyle J={\begin{pmatrix}1&0&0&0&0&0&0\0円&1&0&0&0&0&0\0円&0&1&1&0&0&0\0円&0&0&1&1&0&0\0円&0&0&0&1&0&0\0円&0&0&0&0&1&1\0円&0&0&0&0&0&1\end{pmatrix}},}${\displaystyle J={\begin{pmatrix}1&0&0&0&0&0&0\0円&1&0&0&0&0&0\0円&0&1&1&0&0&0\0円&0&0&1&1&0&0\0円&0&0&0&1&0&0\0円&0&0&0&0&1&1\0円&0&0&0&0&0&1\end{pmatrix}},}

where ${\displaystyle M}${\displaystyle M} is a generalized modal matrix for ${\displaystyle A}${\displaystyle A}, the columns of ${\displaystyle M}${\displaystyle M} are a canonical basis for ${\displaystyle A}${\displaystyle A}, and ${\displaystyle AM=MJ}${\displaystyle AM=MJ}.^[8] Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both ${\displaystyle M}${\displaystyle M} and ${\displaystyle J}${\displaystyle J} may be interchanged, it follows that both ${\displaystyle M}${\displaystyle M} and ${\displaystyle J}${\displaystyle J} are not unique.^[9]

• Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields , Boston: Houghton Mifflin Co., ISBN 0-395-14017-X
• Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646

• Matrices (mathematics)