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Finding the (Diagonalized) Modal Representation

The $ i$ th eigenvector $ \underline{e}_i$ of a matrix $ \mathbf{A}$ has the defining property

$\displaystyle \mathbf{A}\underline{e}_i= {\lambda_i}\underline{e}_i,
$

where $ {\lambda_i}$ is the associated eigenvalue. Thus, the eigenvector $ \underline{e}_i$ is invariant under the linear transformation $ \mathbf{A}$ to within a (generally complex) scale factor $ {\lambda_i}$ .

An $ N\times N$ matrix $ \mathbf{A}$ typically has $ N$ eigenvectors.1Let's make a similarity-transformation matrix $ \mathbf{E}$ out of the $ N$ eigenvectors:

$\displaystyle \mathbf{E}= \left[\begin{array}{cccc} \underline{e}_1 & \underline{e}_2 & \cdots & \underline{e}_N \end{array}\right]
$

Then we have

$\displaystyle \mathbf{A}\mathbf{E}= \left[\begin{array}{cccc} \lambda _1 \underline{e}_1 & \lambda _2\underline{e}_2 & \cdots & \lambda _N\underline{e}_N \end{array}\right]
\isdef \mathbf{E}{\bm \Lambda}
$

where $ {\bm \Lambda}\isdeftext$   diag$ (\underline{\lambda})$ is a diagonal matrix having $ \underline{\lambda}\isdeftext \left[\begin{array}{cccc} \lambda _1 & \lambda _2 & \cdots & \lambda _N \end{array}\right]^T$ along its diagonal. Premultiplying by $ \mathbf{E}^{-1}$ gives

$\displaystyle \zbox{\mathbf{E}^{-1} \mathbf{A}\mathbf{E}= {\bm \Lambda}}
$

Thus, $ \mathbf{E}= \left[\begin{array}{cccc} \underline{e}_1 & \underline{e}_2 & \cdots & \underline{e}_N \end{array}\right]$ is a similarity transformation that diagonalizes the system.


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``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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