Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors were introduced in the previous section. In this section, a minimal discussion will be provided to use them effectively.

Definizione 1   Given a square matrix $\mathbf{A}$ of order $n$, a number (real or complex) $\lambda$, and a non-zero vector $\mathbf {x}$ are said to be an eigenvalue and an eigenvector of $\mathbf{A}$ if the following relation holds:

$$\mathbf{A}\mathbf{x}=\lambda\mathbf{x}$$ (1.10)

$\mathbf {x}$ is also referred to as the eigenvector associated with the eigenvalue $\lambda$.

An eigenvector is a vector $\mathbf{x} \neq 0$ that does not change direction due to the geometric linear transformation (application) $\mathbf{A}$ but only changes its magnitude by a factor $\lambda$ known as the eigenvalue.

Rewriting the system (1.10) using the identity matrix $\mathbf{I}$, it follows that the eigenvalue and its associated eigenvector are obtained as the solution of the homogeneous system:

$$(\mathbf{A} - \lambda\mathbf{I})\mathbf{x}=0$$ (1.11)

If $\mathbf {x}$ is an eigenvector of $\mathbf{A}$ associated with the eigenvalue $\lambda$ and $t \neq 0$ is a number (real or complex), then $t\mathbf{x}$ is also an eigenvector of $\lambda$.

In general, the set of vectors $\mathbf {x}$ associated with an eigenvalue $\lambda$ of $\mathbf{A}$ forms a subspace of $\mathbb{R}^{n}$ called the eigenspace.

The dimension of this subspace is called the geometric multiplicity of the eigenvalue.

Definizione 2   The characteristic polynomial of $\mathbf{A}$ in the variable $x$ is defined as follows:

$$p(x) = \det (\mathbf{A} - x \mathbf{I} )$$ (1.12)

From the definition (1.11), it follows that $\lambda$ is an eigenvalue if and only if $p(\lambda)=0$. The roots of the characteristic polynomial are the eigenvalues of $\mathbf{A}$, and consequently, the characteristic polynomial has a degree equal to the dimension of the matrix. The matrices $2 \times 2$ and $3 \times 3$ have notable characteristic polynomials.

Properties of Eigenvalues and Eigenvectors

• $\mathbf{A}$ and $\mathbf{A}^{\top}$ have the same eigenvalues;
• If $\mathbf{A}$ is nonsingular, and $\lambda$ is one of its eigenvalues, then $\lambda^{-1}$ is an eigenvalue of $\mathbf{A}^{-1}$;
• If $\mathbf{A}$ is orthogonal, then $\vert \lambda \vert=1$;
• $\lambda=0$ is an eigenvalue of $\mathbf{A}$ if and only if $\det(\mathbf{A})=0$;
• The eigenvalues of diagonal and triangular matrices (upper and lower) are the elements of the main diagonal;
• The sum of the diagonal elements is equal to the sum of the eigenvalues, that is, $\trace \mathbf{A} = \sum \lambda_i$;
• The determinant of a matrix is equal to the product of its eigenvalues, that is, $\det \mathbf{A} = \prod \lambda_i$;
• Symmetric matrices have real eigenvalues and orthogonal eigenvectors, thus the corresponding set of unit eigenvectors $\left\{\mathbf{v}_i\right\}$ is a set of orthonormal vectors;
• A square matrix $\mathbf{A}$ can be expressed in terms of the eigenvalue matrix $\mathbf{D}=\diag \left( \lambda_1, \ldots, \lambda_n \right)$ and the eigenvalues $\mathbf{V}$ in the form
  

  $$\mathbf{A} = \mathbf{V} \mathbf{D} \mathbf{V}^{-1}$$ (1.13)

  
  

• A symmetric square matrix $\mathbf{A}$ can be expressed in terms of its real eigenvalues $\lambda_i$ and eigenvalues $\left\{\mathbf{v}_i\right\}$ in the form
  

  $$\mathbf{A} = \mathbf{V} \mathbf{D} \mathbf{V}^{\top} = \sum_{i=1}^{n} \lambda_i \mathbf{v}_i \mathbf{v}^{\top}_i$$ (1.14)

  
  
  known as the spectral decomposition of $\mathbf{A}$.