Eigenvalues and Eigenvectors

Basic Definition

For $A\in M_{n\times n}(\mathbb{R})$, a scalar $\lambda$ is an eigenvalue of $A$ if $A\vec{x}=\lambda \vec{x}$ for some nonzero vector $\vec{x}$. The vector $\vec{x}$ is then called an eigenvector of $A$ corresponding to $\lambda$.

Determinants and Eigenstuff

$\lambda$ is a eigenvalue of $A$ if and only if satisfies the equation $\det (A-\lambda I)=0$ All nonzero solutions of the homogeneous system of equations $(A-\lambda I)\vec{x}=\vec{0}$ are all of the eigenvectors corresponding to $\lambda$.

Characteristic Polynomial

The characteristic polynomial of $A$ is $C_A(\lambda )=\det (A-\lambda I)$

$\lambda$ is an eigenvalue of A if and only if $C_A(\lambda )=0$.

Non-real eigenvalues

A real matrix can have non-real eigenvalues.

Eigenspace

Let $\lambda$ be an eigenvalue of $A\in M_{n\times n}(\mathbb{R})$. The set containing all of the eigenvectors of $A$ corresponding to $\lambda$ together with the zero vector of ${\mathbb{R}}^n$ is called the eigenspace of $A$ corresponding to $\lambda$, and is denoted by $E_{\lambda }(A)$. It follows that $E_{\lambda }(A)=Null(A-\lambda I)$

The algebraic multiplicity $a_{\lambda }$ of $\lambda$ is the number of times $\lambda$ appears as a root of $C_A(\lambda )$. The geometric multiplicity $g_{\lambda }$ of $\lambda$ is the dimension of the eigenspace $E_{\lambda }(A)$.

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