Matrix Function

• categories: Linear algebra, Definition

Definition

A matrix function is a function that operates on matrices, analogous to scalar functions. Given a matrix $A\in {\mathbb{R}}^{n\times n}$ and a scalar function $f(x)$ (e.g., $\exp (x)$, $\sin (x)$, $\log (x)$), the matrix function $f(A)$ is defined in a way consistent with $f(x)$ acting on the eigenvalues or through a power series expansion.

Key Methods for Defining Matrix Functions

1. Power Series Expansion: If $f(x)$ has a convergent power series:

   $$f(x)=\sum _{k=0}^{\infty }{c}_k{x}^k,$$

   then:

   $$f(A)=\sum _{k=0}^{\infty }{c}_k{A}^k,$$

   where $A^k$ is the $k$-th power of the matrix $A$.

   Example: For the exponential function:

   $$\exp (A)=\sum _{k=0}^{\infty }\frac{A^k}{k!}.$$

2. Eigenvalue Decomposition: If $A$ is diagonalizable:

   $$A=V\Lambda V^{-1},$$

   where $\Lambda =\text{diag}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$, then:

   $$f(A)=Vf(\Lambda )V^{-1},$$

   where $f(\Lambda )$ applies $f$ to each eigenvalue:

   $$f(\Lambda )=\text{diag}(f({\lambda }_1),f({\lambda }_2),\dots ,f({\lambda }_n)).$$

3. Jordan Normal Form: If $A$ is not diagonalizable, use its Jordan decomposition:

   $$A=PJ{P}^{-1},$$

   where $J$ is the Jordan form. Then:

   $$f(A)=Pf(J)P^{-1}.$$

   Here, $f(J)$ involves handling blocks corresponding to repeated eigenvalues.

4. Numerical Approximation: For large matrices or matrices without explicit decompositions, numerical methods (e.g., Padé approximants, Krylov subspace methods) are used.

Examples of Matrix Functions

1. Exponential of a Matrix ($\exp (A)$)

The matrix exponential is defined as:

$$\exp (A)=\sum _{k=0}^{\infty }\frac{A^k}{k!}.$$

It is used in solving systems of differential equations:

$${\mathbf{x}}^{\prime }=A\mathbf{x}⟹\mathbf{x}(t)=\exp (At)\mathbf{x}(0).$$

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2. Logarithm of a Matrix ($\log (A)$)

The matrix logarithm is the inverse of the matrix exponential:

$$\log (A)\text{ satisfies }\exp (\log (A))=A.$$

For diagonalizable $A$:

$$\log (A)=V\log (\Lambda )V^{-1}.$$

3. Square Root of a Matrix ($\sqrt{A}$)

The matrix square root satisfies:

$$\sqrt{A}\cdot \sqrt{A}=A.$$

For diagonalizable $A$:

$$\sqrt{A}=V\sqrt{\Lambda }{V}^{-1},$$

where $\sqrt{\Lambda }=\text{diag}(\sqrt{{\lambda }_1},\sqrt{{\lambda }_2},\dots )$.

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4. Trigonometric Matrix Functions

• Sine:

  $$\sin (A)=\sum _{k=0}^{\infty }(-1{)}^k\frac{A^{2k+1}}{(2k+1)!}.$$

• Cosine:

  $$\cos (A)=\sum _{k=0}^{\infty }(-1{)}^k\frac{A^{2k}}{(2k)!}.$$

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Properties

1. Commutativity with Diagonalizable Matrices:
   If $A$ and $B$ commute ($AB=BA$), then:

   $$f(A+B)=f(A)+f(B).$$

2. Similarity Invariance:
   For any invertible matrix $P$:

   $$f(PA{P}^{-1})=Pf(A)P^{-1}.$$

3. Taylor Series Consistency:
   Matrix functions extend scalar Taylor series expansions to matrices.

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Applications

1. Differential Equations:
   Solving linear systems ${\mathbf{x}}^{\prime }=A\mathbf{x}$ involves $\exp (A)$.

2. Quantum Mechanics:
   Time evolution of quantum states uses matrix exponentials.

3. Control Theory:
   Stability analysis uses functions like $\exp (At)$.

4. Data Science and Machine Learning:
   Regularization (e.g., $\log (A)$ or $\sqrt{A}$) in covariance estimation.