Basic properties of eigenvalue problems#

Definition and motivation#

Let \(A\in\mathbb{C}^{n\times n}\). An eigenvalue problem takes the form to find \(x\in\mathbb{C}^n\), \(x\neq 0\) and \(\lambda\in\mathbb{C}\) such that

\[ Ax = \lambda x. \]

Eigenvalue problems arise naturally in the study of initial value problems. Consider the problem of solving the ODE

\[ y' = Ay \]

for \(A\in\mathbb{C}^{n\times n}\) with \(y(0) = y_0\).

By writing \(y(t) = y_0e^{\lambda t}\) and substituting we obtain the eigenvalue problem

\[ Ay_0 = \lambda y_0. \]

When reasoning about eigenvalues it therefore makes sense to think of eigenvalues not just as properties of matrices but as properties of dynamical systems driven by the matrix \(A\). This contrasts singular values, which describe the mapping properties of a matrix \(A\) and are therefore intuitively quite distinct from eigenvalues.

Characteristic polynomials, algebraic and geometric multiplicities#

We can rewrite an eigenvalue problem in the form

\[ (\lambda I - A)x = 0. \]

This has nontrivial solutions \(x\) if and only if \(\lambda I - A\) is singular, or equivalently

\[ \det(\lambda I - A) = 0. \]

We define the characteristic polynomial

\[ p(\lambda) := \det(\lambda I - A). \]

By expanding the determinant we can see that \(p\) is a polynomial of maximum degree \(n\). Denote the \(j\)the zero of \(p\) as \(\lambda_j\). We have that

\[ P(\lambda) = \prod_j (\lambda -\lambda_j)^\alpha_j, \]

where \(\alpha_j\) denotes the algebraic multiplicity of the \(j\)th root. From the fundamental theorem of algebra we know that \(n = \sum_j \alpha_j\). Hence, if we count by multiplicities an \(n\times n\) matrix has \(n\) eigenvalues.

For eigenvalue problems also of relevance is the geometrix multicity \(\beta_j\) of an eigenvalue \(\lambda_j\). \(\beta_j\) is defined as the dimension of the nullspace of

\[ \lambda_j I - A. \]

If \(\lambda_j\) is a single eigenvalue then this dimension is \(1\). If \(\alpha_j >1\) we have that \(\beta_j \leq \alpha_j\). More about this relationship is obtained by discussing the Jordan normal form of a matrix, which is not topic of this module.

The eigenvalue decomposition#

If \(\alpha_j=\beta_j\) for all \(j\) the following eigenvalue decomposition exists

\[ A = X\Lambda X^{-1}, \]

where \(X\in\mathbb{C}^{n\times n}\) is a nonsingular matrix whose columns \(x_j\) are the eigenvectors of \(A\) and \(\Lambda\) is a diagonal matrix whose diagonal entries are the associated eigenvalues \(\lambda_j\).