Eigenvalues and Eigenvectors

Nathan Cooper

June 25, 2018

Eigenvalues and Eigevectors

\[ Ax = \lambda x \]

Where we are looking for the null spaces of A where x is non-zero:

\[ (A - \lambda I)x = 0 \] \[ \boxed{det(A - \lambda I) = 0} \]

The Determinant

For 2x2 and 3x3 matrices Determinant can be found by multiplying factors along the diagonals of a matrix, the adding/subtracting terms on different diagonals:

\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} \]

Factors along the solid lines are added, factors along the dotted lines are subtracted.

This generates a 3\(^{rd}\) degree polynomial. The eigenvalues are the roots.

The eigenvectors are solved by setting \(\lambda\) equal to the root and solving for the vector x.

Example Rotational Axis of an object.

\[ I = m*r^2 \]

\[ \textbf{M} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \\ \end{bmatrix} \]

Example part 2.

A system of particles has an Inertia matrix defined by M in the code below. What are the systems ‘preferred’ axes of rotations and there inertia values?

##      [,1] [,2] [,3]
## [1,]  1.0  0.5  0.0
## [2,]  0.5  1.0  0.5
## [3,]  0.0  0.5  1.0
## [1] 1.7071068 1.0000000 0.2928932
##           [,1]          [,2]       [,3]
## [1,] 0.5000000 -7.071068e-01 -0.5000000
## [2,] 0.7071068 -1.099065e-15  0.7071068
## [3,] 0.5000000  7.071068e-01 -0.5000000

Solutions

\[ \lambda_1 = \sqrt{2} \]

\[ x_1 = \begin{bmatrix} \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \end{bmatrix} \]

\[ \lambda_2 = 1 \]

\[ x_2 = \begin{bmatrix} \frac{-1}{\sqrt{2}} & 0 & \frac{1}{\sqrt2} \\ \end{bmatrix} \]

\[ \lambda_3 = 1 - \frac{1}{\sqrt2} = 0.2928932 \]

\[ x_2 = \begin{bmatrix} \frac{-1}{2} & \frac{1}{\sqrt2} & \frac{-1}{2} \\ \end{bmatrix} \]

Simple Experiment:

Eigenvalues and Eigenvectors in SVD

\[ R = U\Sigma V^T \]

SVD with Sparse Data

Sources