MSR/ressources/Eigenvalues and eigenvectors.md

(online ref)

A linear transformation rotates, stretches or shears the vectors upon which it acts. Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.

Let A be an n*n matrix (the transformation matrix), if for a vector v of length n and a scalar \lambda we have :

Av= \lambda v

Then v is an eigenvector of A, with \lambda being its corresponding eigenvalue.

applications

Vibrations

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes.

m\ddot{x}+c\dot{x}+kx=0

in n dimensions can become :

(\omega^2m+\omega c +k)x=0

which is a quadratic eigenvalue problem

Tensors

The eigenvectors of the moment of inertia tensor define the principle axes of a rigid body.

The stress tensor is symmetric and can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis.