Exponential and Logarithm of Matrices 📂Matrix Algebra

Exponential and Logarithm of Matrices

Definition

Aiming to generalize the exponential function $\exp$ and the logarithmic function $\log$ to matrices.

Matrix Exponential

The generalization of the exponential function to matrices $\exp : \mathbb{C}^{n \times n} \to \mathbb{C}^{n \times n}$ is defined for a matrix $A \in \mathbb{C}^{n \times n}$ as follows: $$ \exp A := \sum_{k=0}^{\infty} \frac{A^{k}}{k!} $$ The expression $\exp A$ can also be simply written as $e^{A}$, and it is called the matrix exponential of $A$.

Matrix Logarithm

The generalization of the logarithmic function to matrices $\log : \mathbb{C}^{n \times n} \to \mathbb{C}^{n \times n}$ is defined such that if there exists a matrix $A$ that satisfies $\exp A = B$ with $B$, then we denote $A$ as $\log B$, and it is called the matrix logarithm of $B$.

Theorem ¹

If two matrices $A, B \in \mathbb{C}^{n \times n}$ satisfy $AB = BA$, then the following holds: $$ \log AB = \log A + \log B - 2 \pi i \mathcal{U} \left( \log A + \log B \right) $$ In particular, for the eigenvalues $\mu_{k}$ of $A$ and the eigenvalues $\nu_{k}$ of $B$, we obtain the following corollary: $$ \log AB = \log A + \log B \iff \forall k = 1 , \cdots , n : \arg \mu_{k} + \arg \nu_{k} \in ( - \pi , \pi ] $$ Here $\arg$ is the argument of a complex number, and $\mathcal{U}$ is the matrix unwinding function defined as follows: $$ \mathcal{U} (A) := {\frac{ A - \log e^{A} }{ 2 \pi i }} \qquad , A \in \mathbb{C}^{n \times n} $$

Explanation

The matrix exponential and logarithm formally extend the exponential and logarithm defined in complex numbers. As seen, the matrix exponential is defined using the power series of the matrix.

In the Space of Hermitian and Positive Definite Matrices

Space of Hermitian and Positive Definite Matrices:
The Hermitian matrix space: The set of all Hermitian matrices of size $n \times n$ is represented as follows. For a scalar field $\mathbb{R}$, $\mathbb{H}_{n}$ is a vector space. $$ \mathbb{H}_{n} := \left\{ A \in \mathbb{C}^{n \times n} : A = A^{\ast} \right\} $$
The set of positive definite matrices: The set of all positive definite matrices of size $n \times n$ is denoted as $\mathbb{P}_{n}$. $\mathbb{P}_{n} \subset \mathbb{H}_{n}$ is a convex cone of $\mathbb{H}_{n}$.

Instead of the space of all matrices, considering the Hermitian matrix space $\mathbb{H}_{n}$ as the domain and the convex cone of positive definite matrices $\mathbb{P}_{n}$ as the codomain, $\exp : \mathbb{H}_{n} \to \mathbb{P}_{n}$ is a bijection, with its inverse being $\log : \mathbb{P}_{n} \to \mathbb{H}_{n}$. Particularly, these are diffeomorphisms.