Eigenvalue

An eigenvalue is a scalar associated with an eigenvector. An eigenvector of a linear transformation is a vector that only scales as a result of being transformed, and the scaling factor is it’s eigenvalue.

Eigenvalues also correspond to the roots of a polynomial, and so can be both real or complex.

Definition §

Eigenvalues of a Matrix ^formula

The eigenvalues of a matrix, $\mathbf{A}$ of size $n\times n$, is given by solving the characteristic equation:

$$\det (\mathbf{A}-\lambda \mathbf{I})=0$$

When we do calculate this determinant, we can represent it as a polynomial:

Characteristic Polynomial ^theorem

Let $\mathbf{A}$ be an $n\times n$ matrix The determinant $\det (\mathbf{A}-\lambda \mathbf{I})$ is also called the characteristic polynomial. The roots of this polynomial return the eigenvalues. This polynomial is of degree $n$

By the fundamental theorem of algebra, we can always find at least one complex eigenvalue for any real or complex $n\times n$ matrix. Further, the sum of the algebraic multiplicities must equal n.

Properties §

Algebraic Multiplicity §

Algebraic Multiplicity of an Eigenvalue

The algebraic multiplicity of an eigenvalue, ${\lambda }_0$, is the power of the factor $(\lambda -{\lambda }_0)$ that appears in the characteristic polynomial.

Note that the sum of the algebraic multiplicities of the eigenvalues is equal must equal $n$:

For example, let the characteristic polynomial be $(\lambda -1{)}^2(\lambda -3)$. Then the algebraic multiplicity of the eigenvalue $1$ is 2, (since we have $(\lambda -1)$ raised to the power of 2). Similarly, the algebraic multiplicity of the eigenvalue $3$ is 1.

Geometric Multiplicity §

Geometric Multiplicity of an Eigenvalue

The geometric multiplicity of an eigenvalue, ${\lambda }_0$ is the number of linearly independent eigenvectors associated with the eigenvalue.

Note that the geometric multiplicity of $\lambda \le$ algebraic multiplicity of $\lambda$.

Relation to trace and determinant §

$$\text{Tr}(\mathbf{A})=\sum _{i=1}^n{\lambda }_i=\text{sum of eigenvalues}$$

$$\det (\mathbf{A})=\prod _{i=1}^n{\lambda }_i=\text{product of eigenvalues}$$

Eigenvalues (Derivation) §

An eigenvector for a matrix is as follows:

This equation can be rewritten:

$$\mathbf{A}\vec{v}=\lambda \vec{v}$$

$$\mathbf{A}\vec{v}-\lambda \vec{v}=0$$

$$\mathbf{A}\vec{v}-\lambda \mathbf{I}\vec{v}=0$$

$$(\mathbf{A}-\lambda \mathbf{I})\vec{v}=0$$

This is a matrix representation of a homogenous linear system. We need $\vec{v}$ to be a non-zero vector, else it is not an eigenvector. Furthermore, $(\mathbf{A}-\lambda \mathbf{I})$ needs to be a singular matrix, else we could multiply the entire equation by it:

$$(\mathbf{A}-\lambda \mathbf{I}{)}^{-1}(\mathbf{A}-\lambda \mathbf{I})\vec{v}=(\mathbf{A}-\lambda \mathbf{I}{)}^{-1}0$$

$$\vec{v}=0$$

which is not what we want. Hence the matrix needs to be singular, which is given by:

$$\det (\mathbf{A}-\lambda \mathbf{I})=0$$

where $\det ()$ is the determinant.

The equation is known as the characteristic equation.