Eigenvalues and Diagonalization Workshop

Definition Let $A$ be a real $n \times n$ matrix and let $F$ be $\mathbb{R}$ or $\mathbb{C}$. An eigenvalue of $A$ over $F$ is a number $\lambda$ in $F$ with $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero $\mathbf{v}$ in $F^n$, and these are exactly the roots of $\chi_A(t) = \det(tI - A)$ that lie in $F$. Over $\mathbb{C}$ the characteristic polynomial always factors into $n$ linear factors, counted with multiplicity, while over $\mathbb{R}$ it may not. The matrix $A$ is diagonalizable over $F$ if $A = PDP^{-1}$ with $P$ invertible and $D$ diagonal, both with entries in $F$, which happens exactly when $F^n$ has a basis of eigenvectors of $A$. Diagonalizable over $\mathbb{R}$ implies diagonalizable over $\mathbb{C}$, but not conversely. Changing the field changes which scalars and vectors are allowed, never the entries of $A$.

Definition Let $A$ be a real $n \times n$ matrix and let $\lambda$ be an eigenvalue of $A$, real or complex. The algebraic multiplicity $\operatorname{am}(\lambda)$ is the multiplicity of $\lambda$ as a root of $\chi_A$, and the geometric multiplicity is

For a nonreal $\lambda$ the null space and the rank are taken over $\mathbb{C}$. These always satisfy $1 \le \operatorname{gm}(\lambda) \le \operatorname{am}(\lambda)$. The matrix is defective if $\operatorname{gm}(\lambda) < \operatorname{am}(\lambda)$ for at least one of its eigenvalues, real or complex. To test a matrix, compute $\operatorname{rank}(A - \lambda I)$ for each repeated eigenvalue. A simple root has both multiplicities equal to $1$ and needs no check. A defective matrix has fewer than $n$ independent eigenvectors, so it is diagonalizable neither over $\mathbb{R}$ nor over $\mathbb{C}$. If every root of $\chi_A$ lies in the field $F$, then $A$ is diagonalizable over $F$ exactly when it is not defective. Without that condition a matrix can fail to be defective and still not be diagonalizable over $F$, as the quarter turn over $\mathbb{R}$ shows.

Definition Let $a$ and $b$ be real numbers with $b \ne 0$, and let

With $r = \sqrt{a^2 + b^2}$, the matrix $C$ rotates the plane through the angle $\theta$ with $\cos\theta = a/r$ and $\sin\theta = b/r$ and scales by $r$. Its characteristic polynomial $\chi_C(t) = (t - a)^2 + b^2$ has no real root, so $C$ has no real eigenvalue and no real eigenvector. Over $\mathbb{C}$ it has the conjugate eigenvalues $a \pm bi$, with conjugate eigenvectors,

The eigenvector matrix with columns $(1, -i)$ and $(1, i)$ has determinant $2i \ne 0$, so $C$ is diagonalizable over $\mathbb{C}$. Both eigenvalues have modulus $r$, the scale factor.

Definition A real $n \times n$ matrix $A$ has a real orthogonal diagonalization

with $Q$ real orthogonal and $D$ real diagonal, if and only if $A$ is symmetric. This is the real spectral theorem together with the fact that $QDQ^{\top}$ is always symmetric. The method is to check $A^{\top} = A$, find the eigenvalues, which are real, find an orthonormal basis of each eigenspace, with Gram-Schmidt inside any eigenspace of dimension $2$ or more, and place these vectors in $Q$ with their eigenvalues in matching order in $D$. Because the columns of $Q$ are orthonormal, $Q^{-1} = Q^{\top}$, so no inverse has to be computed.

Definition Suppose $A$ has an eigenbasis $\mathbf{v}_1, \dots, \mathbf{v}_n$ with $A\mathbf{v}_i = \lambda_i\mathbf{v}_i$, so that $A = PDP^{-1}$ with $P = [\mathbf{v}_1 \ \cdots \ \mathbf{v}_n]$ and $D = \operatorname{diag}(\lambda_1, \dots, \lambda_n)$. For every integer $k \ge 0$, $A^k = PD^kP^{-1}$ with $D^k = \operatorname{diag}(\lambda_1^k, \dots, \lambda_n^k)$. To apply $A^k$ to a vector, write $\mathbf{x} = c_1\mathbf{v}_1 + \cdots + c_n\mathbf{v}_n$, where $\mathbf{c} = P^{-1}\mathbf{x}$, and then

With a real orthonormal eigenbasis $\mathbf{q}_1, \dots, \mathbf{q}_n$ the coefficients are dot products, $c_i = \mathbf{q}_i \cdot \mathbf{x}$. The method needs an eigenbasis, so it does not apply to a defective matrix, and every coefficient must be kept, not only the one with the largest eigenvalue.

Definition The hypothesis of a method is the condition under which its conclusion is guaranteed, and it is checked before the method is used. The methods of this unit need the following.

• Diagonalization $A = PDP^{-1}$ over a field $F$ needs $n$ linearly independent eigenvectors in $F^n$.
• Real orthogonal diagonalization $A = QDQ^{\top}$ needs a real symmetric matrix.
• The eigenvalue test for definiteness needs the symmetric matrix of the form, which for $\mathbf{x}^{\top}M\mathbf{x}$ is $\frac{1}{2}(M + M^{\top})$.
• The power formula $A^k = PD^kP^{-1}$ needs an eigenbasis.

When a hypothesis fails, the conclusion can fail too, and the failure points to what to do instead, such as working over $\mathbb{C}$, computing powers directly or replacing a matrix by its symmetric part.