2.4 Explain the steps for diagonalizing a matrix

Diagonalizing matrices is a key technique in linear algebra. It involves finding a diagonal matrix similar to the original matrix, using eigenvalues and eigenvectors. This process simplifies complex matrix operations and reveals important properties of linear transformations.

Understanding diagonalization connects to the broader concept of linear transformations. It shows how matrices can be decomposed and represented in simpler forms, making it easier to analyze their behavior and solve related problems in various applications.

Diagonalizing matrices

Steps for diagonalization

• Diagonalizing a matrix involves finding a diagonal matrix $D$ that is similar to the original square matrix $A$
  • This means finding an invertible matrix $P$ such that $P^{-1}AP = D$
• The columns of matrix $P$ are the eigenvectors of $A$ and the diagonal entries of $D$ are the corresponding eigenvalues
• For an $n \times n$ matrix to be diagonalizable, it must have $n$ linearly independent eigenvectors
  • Matrices that do not satisfy this are not diagonalizable
• The process of diagonalization involves three main steps:
  1. Find the eigenvalues of the matrix $A$ by solving the characteristic equation $\det(A - \lambda I) = 0$
  2. For each distinct eigenvalue $\lambda$, find its corresponding eigenvectors by solving $(A - \lambda I)x = 0$
  3. Form the matrices $P$ and $D$ using the eigenvectors and eigenvalues, then verify that $P^{-1}AP = D$

Conditions for diagonalizability

• A matrix is diagonalizable if and only if it has a full set of linearly independent eigenvectors
  • For an $n \times n$ matrix, this means having $n$ linearly independent eigenvectors
• Equivalently, a matrix is diagonalizable if and only if the algebraic multiplicity equals the geometric multiplicity for each eigenvalue
  • Algebraic multiplicity: the number of times an eigenvalue appears as a root of the characteristic polynomial
  • Geometric multiplicity: the dimension of the eigenspace (the space of all eigenvectors) for an eigenvalue
• Examples of matrices that are not diagonalizable:
  • Matrices with repeated eigenvalues but not enough linearly independent eigenvectors (defective matrices)
  • Matrices with non-real eigenvalues (e.g., rotation matrices)

Eigenvalues and eigenvectors

Finding eigenvalues

• Eigenvalues $\lambda$ are scalar values that satisfy the equation $Ax = \lambda x$ for some nonzero vector $x$
  • The vector $x$ is called an eigenvector corresponding to $\lambda$
• To find the eigenvalues, set up and solve the characteristic equation $\det(A - \lambda I) = 0$
  • This will give a polynomial in $\lambda$
  • The roots of this polynomial (found by setting the equation equal to zero and solving) are the eigenvalues of $A$
• The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial
  • For example, if the characteristic polynomial is $(\lambda - 2)^2(\lambda - 3)$, then the eigenvalue 2 has algebraic multiplicity 2 and the eigenvalue 3 has algebraic multiplicity 1

Finding eigenvectors

• For each distinct eigenvalue $\lambda$, find the corresponding eigenvectors by solving $(A - \lambda I)x = 0$
  • This is equivalent to finding the nullspace of $(A - \lambda I)$
  • Non-trivial solutions to this system are the eigenvectors
• The geometric multiplicity of an eigenvalue is the dimension of its eigenspace (the space of all its eigenvectors)
  • For example, if the eigenspace for $\lambda = 2$ is spanned by two linearly independent vectors, then the geometric multiplicity of 2 is 2
• If the algebraic multiplicity of an eigenvalue is greater than 1, then the corresponding eigenvectors can be chosen to be any linearly independent set that spans the eigenspace

Eigenvalue and eigenvector matrices

Eigenvalue matrix

• The eigenvalue matrix $D$ is a diagonal matrix constructed using the eigenvalues of $A$
  • The eigenvalues are placed on the main diagonal of $D$ in any order
  • If an eigenvalue has algebraic multiplicity greater than 1, it will appear on the diagonal multiple times
  • All off-diagonal entries of $D$ are zero
• For example, if the eigenvalues of $A$ are 2 (with multiplicity 2) and 3, then a possible eigenvalue matrix is: $D = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$

Eigenvector matrix

• The eigenvector matrix $P$ is formed by using the eigenvectors of $A$ as its columns
  • The eigenvectors should be in the same order as their corresponding eigenvalues in $D$
  • Each column of $P$ is an eigenvector $x_i$ that satisfies $Ax_i = \lambda_i x_i$, where $\lambda_i$ is the $i$th eigenvalue in the diagonal of $D$
• If an eigenvalue has algebraic multiplicity greater than 1, then the corresponding eigenvectors can be chosen to be any linearly independent set that spans the eigenspace
• $P$ is invertible if and only if $A$ is diagonalizable
  • If $P$ is invertible, then $P^{-1}AP = D$

Matrix product representation

Diagonalization as matrix factorization

• If $A$ is diagonalizable, it can be factored as $A = PDP^{-1}$, where $D$ is the eigenvalue matrix and $P$ is the eigenvector matrix
  • This factorization is not unique, as the eigenvalues and eigenvectors can be ordered differently in $D$ and $P$
  • If an eigenvalue has algebraic multiplicity greater than 1, then there are infinitely many choices for $P$, but the factorization still holds
• The columns of $P$ form a basis of $\mathbb{R}^n$ consisting of eigenvectors of $A$
  • The factorization $A = PDP^{-1}$ can be viewed as a change of basis to one in which the transformation $A$ is represented by a diagonal matrix

Applications of diagonalization

• Diagonalization can simplify calculations involving powers of $A$
  • $A^k = (PDP^{-1})^k = PD^kP^{-1}$, and powers of a diagonal matrix are easy to compute
  • For example, if $D = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$, then $D^5 = \begin{bmatrix} 2^5 & 0 \\ 0 & 3^5 \end{bmatrix} = \begin{bmatrix} 32 & 0 \\ 0 & 243 \end{bmatrix}$
• Diagonalization can be used to solve systems of linear differential equations
  • If $\mathbf{x}'(t) = A\mathbf{x}(t)$ and $A$ is diagonalizable, then the solution is $\mathbf{x}(t) = Pe^{Dt}P^{-1}\mathbf{x}(0)$, where $e^{Dt}$ is a diagonal matrix with $e^{\lambda_i t}$ on the diagonal