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?

  • $\begingroup$ Are you asking about eigenvalues/eigenvectors of a <s>signal</s> matrix or a <s>function</s> kernel function? $\endgroup$ – Atul Ingle Feb 20 '14 at 20:23
  • $\begingroup$ EigenValues and EigenVectors are properties of systems or transforms such as linear mapping matrices, signals do not have eigenvalues or eigenvectors. Functions considered as signals also do not possess them, however functions considerd as mappings (therefore as transforms of some sort) may possess eigenvalues and eigenvectors $\endgroup$ – Fat32 May 3 '17 at 14:25

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.

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.

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

  • $\begingroup$ no problem. Also, do you mean "linear time invariant system" instead of "linear transform invariant system"? $\endgroup$ – Atul Ingle Feb 20 '14 at 20:44
  • $\begingroup$ @AtulIngle yes, and corrected.. thx $\endgroup$ – lennon310 Feb 20 '14 at 20:45
  • $\begingroup$ @lennon310 thank you for your answer but i literally did not get what u want to say through the above equations.can u please elaborate it . i just want to know relation of signal and eigen values and eigen vectors. $\endgroup$ – Amit_DSP Feb 21 '14 at 9:52
  • $\begingroup$ @lennon310; a +1 for both the Q and A. Did you for how long I've bore that question in my mind! thanks. $\endgroup$ – MimSaad May 3 '17 at 17:26

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