Matrix Functions — Mathematics Knowledge System

Overview

A matrix function f(A) applies a scalar function f to a square matrix A in a way that is consistent with the algebraic structure of A. For diagonalizable matrices, f(A) = P f(D) P^{-1} where f is applied entry-wise to the diagonal. For general matrices, Jordan form or the Cauchy integral formula provides the definition. Matrix functions include matrix square roots, logarithms, sine, cosine, and the exponential.

Intuition

If A = PDP^{-1} with D = diag(λ_1,...,λ_n), then A acts on the eigenbasis by scaling each direction by λ_i. Applying f to A means: go to the eigenbasis, apply f to each eigenvalue, come back. For non-diagonalizable matrices, the Jordan form shows how f and its derivatives contribute near each eigenvalue — each Jordan block of size k requires f, f', ..., f^{(k-1)} evaluated at the eigenvalue.

Formal Definition

Definition

For a diagonalizable matrix A = PDP^{-1} and a scalar function f defined on the spectrum of A:

f(A) = P\, f(D)\, P^{-1}, \quad f(D) = \operatorname{diag}(f(\lambda_1),\ldots,f(\lambda_n))

Diagonalizable case

f(J_k(\lambda)) = \begin{pmatrix} f(\lambda) & f'(\lambda) & \frac{f''(\lambda)}{2!} & \cdots & \frac{f^{(k-1)}(\lambda)}{(k-1)!} \\ & f(\lambda) & f'(\lambda) & \cdots & \vdots \\ & & \ddots & \ddots & \vdots \\ & & & f(\lambda) & f'(\lambda) \\ & & & & f(\lambda) \end{pmatrix}

For a Jordan block of size k, f(J_k(λ)) involves f and its first k-1 derivatives.

Jordan block case

f(A) = \frac{1}{2\pi i} \oint_\Gamma f(z)(zI - A)^{-1}\, dz

Cauchy integral definition (general)

Notation

Notation	Meaning
$f(A)$	Matrix function: application of scalar function f to matrix A
$A^{1/2}$	Matrix square root (when it exists)
$\log(A)$	Matrix logarithm (when it exists)

Properties

Spectral mapping theorem

\text{The eigenvalues of } f(A) \text{ are } f(\lambda_1),\ldots,f(\lambda_n) \text{ where } \lambda_i \text{ are eigenvalues of } A.

Commutativity with similar matrices

f(PAP^{-1}) = P f(A) P^{-1}.

Consistency with scalars

\text{If } A = \lambda I, \text{ then } f(A) = f(\lambda)I.

Matrix square root

\text{Every positive definite matrix has a unique positive definite square root.}

Applications

Real-world · Engineering

Matrix square roots appear in control theory (Riccati equations) and in computing the geometric mean of positive definite matrices for diffusion tensor imaging.

Worked Examples

1
For diagonal matrices, f(A) is diagonal with f applied to each entry.
$A^{1/2} = \begin{pmatrix}\sqrt{4}&0\\0&\sqrt{9}\end{pmatrix} = \begin{pmatrix}2&0\\0&3\end{pmatrix}$
2
Verify: (A^{1/2})^2 = [[4,0],[0,9]] = A. ✓

✓ Answer

A^{1/2} = diag(2, 3).

Practice Problems

Mediumfree response

Let A = [[2,0],[0,3]]. Compute log(A) (matrix logarithm).

Mediumfree response

State the spectral mapping theorem and use it to find the eigenvalues of e^A where A has eigenvalues 1, 2, 3.

Common Mistakes

Common Mistake

f(A) is computed by applying f entry-wise: (f(A))_{ij} = f(a_{ij}).

Matrix functions are NOT entry-wise (except for diagonal matrices). f(A) is defined via eigendecomposition or Jordan form. Entry-wise application gives a different (generally unrelated) matrix.

Quiz

Summary

f(A) for diagonalizable A = PDP^{-1}: apply f to each eigenvalue, then f(A) = P diag(f(λ_i)) P^{-1}.
For a Jordan block J_k(λ): f(J_k(λ)) involves f and its first k-1 derivatives evaluated at λ.
Spectral mapping theorem: eigenvalues of f(A) are {f(λ) : λ eigenvalue of A}.
NOT entry-wise application: f(A)_{ij} ≠ f(a_{ij}) in general.
Examples: matrix exponential, square root, logarithm, sine, cosine.

References

WebsiteWikipedia — Matrix function