# Eigenvalues and Eigenvectors §

### Basic Definition §

For , a scalar is an eigenvalue of if for some nonzero vector . The vector is then called an eigenvector of corresponding to .

### Determinants and Eigenstuff §

is a eigenvalue of if and only if satisfies the equation All nonzero solutions of the homogeneous system of equations are all of the eigenvectors corresponding to .

### Characteristic Polynomial §

The characteristic polynomial of is

is an eigenvalue of A if and only if .

Non-real eigenvalues

A real matrix can have non-real eigenvalues.

### Eigenspace §

Let be an eigenvalue of . The set containing all of the eigenvectors of corresponding to together with the zero vector of is called the eigenspace of corresponding to , and is denoted by . It follows that

The algebraic multiplicity of is the number of times appears as a root of . The geometric multiplicity of is the dimension of the eigenspace .

### Applications §

• [[notes/Matrix#Powers of Matrices|Matrix#Powers of Matrices]]

Is this stuff really needed?

For a while, I saw no practical application of this. But it turns out that this kind of stuff is super important in Robotics. For instance, the usage of the Jacobian Matrix for robot control, the eigenvalues of this matrix can be calculated to understand the manipulability of the robot.

• I don’t understand what this means