# Why do we need the eigenvalue? [closed]

Now I'm trying to review about the image noise. http://people.csail.mit.edu/celiu/denoise/estnoise/and I have found an article about the eigenvalue.

Why do we need the eigenvalue, what is its use?

• I doubt anyone is going to take the time to write a primer on eigenvectors - it's a big topic. You need to take a course or read a textbook on linear algebra. – Brionius Apr 29 '15 at 0:42

## 1 Answer

All finite dimensional linear transformations leave the direction of at least one vector fixed: They only scale that vector but leave it pointing in the same direction. The scaling constant is the eigenvalue.

In physics, eigenvalues and vectors are mainly meaningful as:

1. Basis vectors for vector spaces that are left invariant by transformations. They are all about invariant spaces;
2. A means of decomposing a linear system, which often completely resolves into eigenvectors through the Spectral Theorem so that the knowledge of a system's eigenvectors completely characterises the system: all other states are superpositions of eigenvectors and linearity by definition means that the sum of transformed eigenvectors is the same as the tranformation of the sum (superposition). This is especially useful in quantum mechanics. Eigenvalues in quantum mechanics are possible measurements returned by a quantum observable.

In image processing, the maximum noise boost imparted by a linear transformation happens when the image is the eigenvector with the maximum magnitude eigenvalue $\lambda_{max}$. The noise power boost is a factor of $\sigma_{max}=|\lambda_{max}|^2$ - the maximum singular value. The mininum noise power boost is likewise $\sigma_{min}=|\lambda_{min}|^2$ the square of the magnitude of the minimum magnitude eigenvalue. The ratio of the two: $\sigma_{max}/\sigma_{min}$ is the condition number and it determines how numerically noisy the inversion of the transformation is (e.g. to infer untransformed data from transformed measurements). For a near-singular matrix, the condition number becomes huge, reflecting the high roundoff error inherent in an attempted inversion.

• Nice compact explanation! – CuriousOne Apr 29 '15 at 1:13
• Thank you Sir, but How do we get the maximum singular value and eigen value from the noisy image to practically calculation. My goal is make the noise curve like the paper. – gmotree Apr 29 '15 at 3:58
• @gmotree The eigenvectors and values pertain to linear transformations on the image, not the image itself. For example, suppose you know that the image is filtered in some way by hardware. You may want to do a deconvolution to see what it would be like were the hardware not in the way. A shift-invariant transformation is diagonalized, i.e. decomposed into its expansion of operators by the Fourier transform and therefore the maximum singular value is the square modulus of the peak frequency response, the minimum SV the corresponding square modulus for the minimum frequency respose.... – Selene Routley Apr 29 '15 at 4:04
• @gmotree ..... Since the latter is often nought, the transformation has destroyed information about the image at the zero frequency response points. So you need to calculate some variation on a Penrose inverse to deconvolve without adding too much noise. The eigenvectors in this case are images with sinusoidal variations of intensity with position. – Selene Routley Apr 29 '15 at 4:07