What are Hermitian and skew-Hermitian matrices and what properties do they have?

In one line

A complex square matrix is Hermitian if it equals its conjugate transpose (\(A^\theta=A\)) and skew-Hermitian if \(A^\theta=-A\); they are the complex versions of symmetric and skew-symmetric matrices.

Explained simply

For real matrices, symmetry means "same in the diagonal mirror". For complex entries, the mirror also changes \(i\) to \(-i\) (takes the conjugate). So a Hermitian matrix has a real diagonal and each entry below the diagonal is the conjugate of the matching entry above. In quantum physics, every measurable quantity (energy, momentum) is represented by a Hermitian matrix, because its eigenvalues are always real.

Technically, \(A^\theta=(\bar A)^T\) is the conjugate transpose (also written \(A^*\) or \(A^H\)). \(A\) is Hermitian if \(a_{ij}=\overline{a_{ji}}\) and skew-Hermitian if \(a_{ij}=-\overline{a_{ji}}\).

How it works

1. Take the conjugate of every entry (change \(i\) to \(-i\)).
2. Transpose to get \(A^\theta\).
3. Compare: \(A^\theta=A\) means Hermitian; \(A^\theta=-A\) means skew-Hermitian.
4. To split any square matrix: \(P=\frac12(A+A^\theta)\) is Hermitian, \(Q=\frac12(A-A^\theta)\) is skew-Hermitian, and \(A=P+Q\).

Key formula

• Diagonal entries of a Hermitian matrix are real; of a skew-Hermitian matrix are purely imaginary or zero.
• If \(A\) is Hermitian, \(iA\) is skew-Hermitian (and vice versa).
• \((A^\theta)^\theta=A\), \((AB)^\theta=B^\theta A^\theta\), \((kA)^\theta=\bar kA^\theta\).
• Eigenvalues of a Hermitian matrix are real; of a skew-Hermitian matrix are purely imaginary or zero.

Quick example

\(A=\begin{bmatrix}2&3+i\\3-i&5\end{bmatrix}\): conjugate \(\begin{bmatrix}2&3-i\\3+i&5\end{bmatrix}\), transpose gives back \(A\). Hermitian.

Split \(B=\begin{bmatrix}1&2i\\3&1+i\end{bmatrix}\): \(B^\theta=\begin{bmatrix}1&3\\-2i&1-i\end{bmatrix}\).

\(P=\frac12(B+B^\theta)=\begin{bmatrix}1&\frac{3+2i}2\\\frac{3-2i}2&1\end{bmatrix}\) (Hermitian), \(\ Q=\frac12(B-B^\theta)=\begin{bmatrix}0&\frac{-3+2i}2\\\frac{3+2i}2&i\end{bmatrix}\) (skew-Hermitian). \(P+Q=B\).

Common mistakes

• Transposing without conjugating (that only tests symmetry).
• Expecting real off-diagonal entries; only the diagonal of a Hermitian matrix must be real.
• Writing \((AB)^\theta=A^\theta B^\theta\); the order reverses.

How MU asks it

Asked as 3-mark or 5-mark parts: "Express \(A\) as \(P+iQ\) where \(P\) and \(Q\) are both Hermitian" (take \(P=\frac12(A+A^\theta)\), \(Q=\frac1{2i}(A-A^\theta)\)), or "Show that \(A\) is Hermitian and \(iA\) is skew-Hermitian". Properties are without proof in the syllabus, so quote them, then compute \(A^\theta\) carefully and verify.