eigen values and eigen vectors of signal

Question

What does the Eigen values and Eigen vector of a signal or function represent? What is its physical significance? I know about basis vectors of a signal which constitute the orthogonal planes where signal projections are represented. Are basis vectors and Eigen vectors same thing? Can we reconstruct signal using these Eigen vectors?

Answer 1

Consider a linear time-invariant system that maps a given signal to another signal space. If the system produces a scaled version of the input signal $\phi$, say $\lambda \phi$, then we can view $\lambda$ and $\phi$ as eigenvalue and eigenvector respectively ($λ$ gives us the gain or attenuation of the eigen-signal).

Now suppose the impulse response of system is $h[n]$, when you input $x[n]$ is an eigen-signal, you have the output $$y[n] = x[n]\sum\limits_{k=- \infty}^{\infty} h[k]e^{-j \cdot \omega \cdot k}$$

so $$\lambda = \sum\limits_{k=- \infty}^{\infty} h[k]e^{-j \cdot \omega \cdot k}$$

Note this is just the Discrete Time Fourier Transform of $h[n]$ since $H(e^{j \omega}) = \sum\limits_{k=- \infty}^{\infty} h[k]e^{-j \cdot \omega \cdot k}$. Further, the Fourier Transform of $x[n]$ becomes meaningful as well.

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Note that eigenvectors do not always form a basis. For example, $\begin{pmatrix} 0 &1 \\ 0 &0\end{pmatrix}$ has $0$ as its only eigenvalue, with eigenspace $\begin{pmatrix} x \\ 0 \end{pmatrix}$. There are not enough independent eigenvectors to form a basis.

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For other discussions on the physical significance of eigenvalues or eigenvectors for a signal, please refer to this post from researchgate. And yes you can reconstruct the original signal using all the eigenvectors, or approximate the signal using some of them