Matrix Functions and Exponentials

A matrix function is an operator built by applying a function to a matrix or finite-dimensional operator in a basis-independent way. The most important example in quantum mechanics is the exponential, which turns a Hamiltonian into a time-evolution operator.

This page treats the finite-dimensional mathematical construction. The Core Formalism interpretation for observables and spectra is Functions of Operators. Numerical algorithms for computing or applying exponentials are collected in Matrix Exponentials Numerically.

Polynomial Functions

The safest starting point is a polynomial. If

$$p(z) = a_0+a_1z+\cdots+a_mz^m,$$

then

$$p(A) = a_0I+a_1A+\cdots+a_mA^m.$$

This definition uses addition, scalar multiplication, and composition of the operator with itself. It is independent of the basis used to write the matrix.

For example,

$$p(A)=A^2-3A+2I$$

means

$$A\circ A-3A+2I,$$

not an entrywise operation on a displayed array.

Power Series Definition

If a function has a power series

$$f(z) = \sum_{k=0}^{\infty} c_k z^k,$$

one defines

$$f(A) = \sum_{k=0}^{\infty} c_k A^k$$

whenever the matrix series converges. For finite matrices, the usual entire functions such as $\exp z$, $\sin z$, and $\cos z$ are defined for every matrix because their power series converge for all inputs.

The matrix exponential is

$$e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}.$$

It is always invertible, with inverse

$$\left(e^A\right)^{-1}=e^{-A}.$$

Diagonalizable Matrices

If $A$ is diagonalizable,

$$A=PDP^{-1},$$

where

$$D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n),$$

then

$$f(A) = P f(D) P^{-1},$$

with

$$f(D) = \operatorname{diag} \left( f(\lambda_1),\ldots,f(\lambda_n) \right).$$

The eigenvectors are unchanged; the eigenvalues are transformed by $f$.

For the exponential,

$$e^A = P \operatorname{diag} \left( e^{\lambda_1},\ldots,e^{\lambda_n} \right) P^{-1}.$$

If $A$ is normal, the diagonalization can be chosen unitary:

$$A=UDU^\dagger, \qquad f(A)=Uf(D)U^\dagger.$$

Spectral Projector Form

For a finite-dimensional Hermitian operator,

$$A = \sum_a aP_a,$$

where $P_a$ is the orthogonal projector onto the eigenspace for eigenvalue $a$. Then

$$f(A) = \sum_a f(a)P_a.$$

This form handles degeneracy cleanly because each projector represents an entire eigenspace. It is the finite-dimensional functional calculus used for observables.

For the projector algebra behind this expression, see Spectral Decomposition.

Matrix Exponentials and Time Evolution

If $H$ is a finite-dimensional Hermitian Hamiltonian, then

$$U(t) = \exp(-iHt/\hbar)$$

is unitary. In a spectral decomposition

$$H = \sum_E E P_E,$$

the exponential is

$$U(t) = \sum_E e^{-iEt/\hbar}P_E.$$

Each energy eigenspace receives a phase. Since

$$\left\lvert e^{-iEt/\hbar}\right\rvert=1,$$

the operator preserves norms and inner products.

The dynamical interpretation is developed in Unitary Time Evolution and Time-Evolution Operator.

Exponential of Pauli Matrices

Let

$$\sigma_{\mathbf n} = n_x\sigma_x+n_y\sigma_y+n_z\sigma_z,$$

where $\mathbf n$ is a real unit vector and $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices. The Pauli algebra gives

$$\sigma_{\mathbf n}^2=I.$$

Split the exponential power series into even and odd powers:

$$\exp \left( -i\frac{\theta}{2}\sigma_{\mathbf n} \right) = \cos\frac{\theta}{2}\,I -i\sin\frac{\theta}{2}\,\sigma_{\mathbf n}.$$

This compact identity is the finite-dimensional engine behind spin-$1/2$ rotations and many single-qubit gates. The Pauli algebra is summarized in Pauli Matrices.

For the special case $\sigma_z$,

$$\exp \left( -i\frac{\theta}{2}\sigma_z \right) = \begin{pmatrix} e^{-i\theta/2} & 0\\ 0 & e^{i\theta/2} \end{pmatrix}.$$

When Exponentials Multiply

If $A$ and $B$ commute, then

$$e^{A+B} = e^A e^B = e^B e^A.$$

If $A$ and $B$ do not commute, this identity generally fails. The first correction is controlled by commutators, and the dynamics story leads to Baker–Campbell–Hausdorff Formula and time-ordering formulas.

This is why time-dependent Hamiltonians are subtle. In general,

$$\exp \left( -\frac{i}{\hbar} \int_{t_0}^t H(s)\,ds \right)$$

is the correct time-evolution operator only when the Hamiltonians at different times commute with one another, or under special conditions that reduce the time-ordered exponential to an ordinary exponential.

Non-Diagonalizable Matrices

Diagonalization is the cleanest route, but the power-series definition also works for non-diagonalizable matrices. If

$$J = \lambda I+N, \qquad N^m=0,$$

then

$$e^J = e^\lambda \left( I+N+\frac{N^2}{2!} +\cdots+ \frac{N^{m-1}}{(m-1)!} \right).$$

The nilpotent part $N$ contributes polynomial factors. This is one reason non-diagonalizable operators behave differently from normal operators, even when their eigenvalues look simple.

Common Mistakes

• Applying a scalar function entry by entry to a matrix in an arbitrary basis.
• Forgetting the identity term $a_0I$ in a polynomial $p(A)$.
• Assuming $e^{A+B}=e^Ae^B$ without checking $AB=BA$.
• Treating diagonalization formulas as valid for non-diagonalizable matrices.
• Forgetting that $e^{-iHt/\hbar}$ is unitary only when $H$ is Hermitian or self-adjoint in the relevant setting.
• Ignoring branch choices for multivalued functions such as logarithms and fractional powers.
• Applying finite-dimensional formulas to unbounded operators without checking domains.

Cross-Links

• Matrices as Linear Maps
• Diagonalization
• Normal Operators
• Hermitian Operators
• Unitary Operators
• Matrix Exponentials Numerically
• Spectral Decomposition
• Commutators and Anticommutators
• Pauli Matrices
• Functions of Operators
• Unitary Time Evolution
• Time-Dependent Hamiltonians

References

• N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008.
• R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013.
• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Let

$$A = \begin{pmatrix} 2 & 0\\ 0 & 5 \end{pmatrix}.$$

Compute $e^A$.

Solution

Since $A$ is diagonal,

$$e^A = \begin{pmatrix} e^2 & 0\\ 0 & e^5 \end{pmatrix}.$$

2. Suppose $A=PDP^{-1}$ with $D=\operatorname{diag}(1,3)$. Write $e^A$.

Solution

Use the diagonalizable-matrix formula:

$$e^A = P \begin{pmatrix} e & 0\\ 0 & e^3 \end{pmatrix} P^{-1}.$$

3. Use $\sigma_x^2=I$ to compute $\exp(-i\theta\sigma_x/2)$.

Solution

Split the power series into even and odd powers:

$$\exp \left( -i\frac{\theta}{2}\sigma_x \right) = \cos\frac{\theta}{2}\,I -i\sin\frac{\theta}{2}\,\sigma_x.$$

4. Why is $e^{A+B}=e^Ae^B$ safe when $AB=BA$ but not in general?

Solution

When $A$ and $B$ commute, products in the power series can be rearranged as in the scalar binomial theorem, so the exponential of the sum factors. When they do not commute, terms such as $AB$ and $BA$ cannot be combined freely, and commutator corrections appear.