Introduction to Quantum Mechanics · Problem A.31

Afterword — David J. Griffiths

Physics — Quantum Mechanics · hermitian-transformations

Problem

Functions of matrices are typically defined by their Taylor series expansions. For example,

e^{\mathsf{M}} \equiv \mathsf{I} + \mathsf{M} + \frac{1}{2}\mathsf{M}^2 + \frac{1}{3!}\mathsf{M}^3 + \cdots.

(a) Find

\exp(\mathsf{M})

, if

(\text{i})\; \mathsf{M} = \begin{pmatrix} 0 & 1 & 3 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{pmatrix}; \qquad (\text{ii})\; \mathsf{M} = \begin{pmatrix} 0 & \theta \\ -\theta & 0 \end{pmatrix}.

(b) Show that if

\mathsf{M}

is diagonalizable, then

\det\left(e^{\mathsf{M}}\right) = e^{\text{Tr}(\mathsf{M})}.

Comment: This is actually true even if

\mathsf{M}

is not diagonalizable, but it's harder to prove in the general case. (c) Show that if the matrices

\mathsf{M}

and

\mathsf{N}

commute, then

e^{\mathsf{M}+\mathsf{N}} = e^{\mathsf{M}} e^{\mathsf{N}}.

Prove (with the simplest counterexample you can think up) that this equation is not true, in general, for non-commuting matrices. (d) If

\mathsf{H}

is hermitian, show that

e^{i\mathsf{H}}

is unitary.

Intuitive explanation

Matrix exponentials extend the familiar scalar $e^x$ to matrices via their Taylor series, but matrix multiplication doesn't commute—so the algebra works differently. The key insight is that many properties of scalar exponentials (like $\det(e^M) = e^{\text{Tr}(M)}$ or $e^{M+N} = e^M e^N$ ) survive only under special conditions: diagonalizability, commutation, or hermiticity. Understanding when and why these familiar rules break down will deepen your grasp of how matrix structure constrains what's true.

Intuitive explanation

Related theory

Hermitian Skew Hermitian And Unitary Matrices

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