12 Determinants and Eigensystems
Determinants
For a square matrix \mathbf{A} \in \mathbb{C}^{n \times n}, the determinant of \mathbf{A} is defined to be the scalar
\text{det} (\mathbf{A}) = \lvert \mathbf{A} \rvert = \sum_{p \in \mathcal{P}} \sigma (p) \prod_{i=1}^{n} a_{i, p_{i}}
where
\mathcal{P} is the set of all permutations of the set of (1, 2, \dots, n),
\sigma (p) is the parity of the permutation p.
Note that each term a1p1a2p2 · · · anpn in (6.1.1) contains exactly one entry from each row and each column of A.
Eigensystems
Definition 12.1 For a square matrix \mathbf{A} \in \mathbb{C}^{n \times n}, (\lambda, \mathbf{x}) is an eigenpair for \mathbf{A} if
\mathbf{A} \mathbf{x} = \lambda \mathbf{x}
where
the scalar \lambda is an eigenvalue for \mathbf{A},
and the non-zero vector \mathbf{x} is an eigenvector associated with \lambda for \mathbf{A}.
Definition 12.2 The set of all distinct eigenvalues, denoted by \sigma (\mathbf{A}), is the spectrum of \mathbf{A}.
Eigenspace
Definition 12.3 Given a eigenvalue \lambda, the eigenspace of \lambda is defined as
N (\mathbf{A} - \lambda \mathbf{I}),
which contains the set of all eigenvectors associated with \lambda
\begin{aligned} \mathbf{A} \mathbf{x} & = \lambda \mathbf{x} \\ (\mathbf{A} - \lambda \mathbf{I}) \mathbf{x} & = 0 \\ \mathbf{x} & \in N (\mathbf{A} - \lambda \mathbf{I}). \\ \end{aligned}
Because of \mathbf{x} \neq \mathbf{0} and rank-nullity theorem,
N (\mathbf{A} - \lambda \mathbf{I}) \neq \{ \mathbf{0} \} \Rightarrow \text{rank} (\mathbf{A} - \lambda \mathbf{I}) < n
which, according to the property of rank, indicates that \mathbf{A} - \lambda \mathbf{I} is a singular matrix, that is,
\text{det} (\mathbf{A} - \lambda \mathbf{I}) = 0.
Characteristic polynomial and equation
Definition 12.4 The characteristic polynomial of a square matrix \mathbf{A} \in \mathbb{C}^{n \times n} is a polynomial function
p (\lambda) = \text{det} (\mathbf{A} - \lambda \mathbf{I}).
The degree (highest exponent in the expression) of p (\lambda) is n,
and the leading term in p (\lambda) is (-1)^{n} \lambda^{n}.
Definition 12.5 The characteristic equation for \mathbf{A} is
p (\lambda) = 0.
The eigenvalues of \mathbf{A} are the solutions of the characteristic equation.
Thus, \mathbf{A} has n eigenvalues, but some may be complex numbers (even if the entries of \mathbf{A} are real numbers), and some eigenvalues may be repeated.
Multiplicities
Definition 12.6 Let \lambda_{1}, \dots, \lambda_{n} \in \sigma (\mathbf{A}) be the spectrum of \mathbf{A}. Then, we define the following multiplicities of an eigenvalue.
The algebraic multiplicity of an eigenvalue \lambda_{i} is the number of times it appears as the root of p (\lambda),
\text{alg mult}_{\mathbf{A}} (\lambda_{i}) = a_{i} \iff \dots + (\lambda - \lambda_{i})^{a_{i}} + \dots = 0.
That is, the algebraic multiplicity of \lambda_{i} is the number of times \lambda_{i} has repeated in all eigenvalues.
The geometric multiplicity of an eigenvalue \lambda_{i} is the dimension of eigenspace associated with \lambda_{i},
\text{geo mult}_{\mathbf{A}} (\lambda_{i}) = \text{dim} N (\mathbf{A} - \lambda_{i} \mathbf{I}).
That is, the geometric multiplicity of \lambda_{i} is the number of linearly independent eigenvectors associated with \lambda_{i}.
For every \mathbf{A} \in \mathbb{C}^{n \times n}, and for each \lambda_{i} \in \sigma(\mathbf{A}),
\text{geo mult}_{\mathbf{A}} (\lambda_{i}) \leq \text{alg mult}_{\mathbf{A}} (\lambda_{i}).
Special multiplicities
When \text{alg mult}_{\mathbf{A}} (\lambda_{i}) = 1, \lambda_{i} is called a simple eigenvalue, since there can only be one unique eigenvector associated with this eigenvalue.
Eigenvalues such that \text{alg mult}_{\mathbf{A}} (\lambda_{i}) = \text{geo mult}_{\mathbf{A}} (\lambda_{i}) are called semi-simple eigenvalues of A, as all eigenvectors associated with the eigenvalues that have the value of \lambda_{i} are linearly independent.
Independent eigenvectors
Let \{ \lambda_{1}, \dots, \lambda_{k} \} be a set of distinct eigenvalues for \mathbf{A}.
Theorem 12.1 If \{ (\lambda_{1}, \mathbf{x}_{1}), \dots, (\lambda_{k}, \mathbf{x}_{k}) \} is a set of eigenpairs for \mathbf{A}, then \{ \mathbf{x}_{1}, \dots, \mathbf{x}_{k} \} is a linearly independent set. That is, the eigenvectors associated with different eigenvalues
Theorem 12.2 If \mathcal{B}_{i} is a basis for eigenspace N (\mathbf{A} - \lambda_{i} \mathbf{I}) of \lambda_{i}, then the union of all basis vectors \mathcal{B} = \mathcal{B}_{1} \cup \dots, \mathcal{B}_{k} for all eigenspace is a linearly independent set.