Last modified: 2026-10-27 20:53:54 (PDT)
Definition 1 (Complex vector) A complex vector of length \(p\) is a column vector whose entries are complex numbers; the set of all of them is denoted \(\mathbb{C}^p\).
Definition 2 (Complex matrix) A complex matrix of size \(m \times n\) is a matrix with complex entries; the set of all of them is denoted \(\mathbb{C}^{m \times n}\).
Example 1 (A complex vector and its conjugate) \[ \underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}_{3 \times 1} = \begin{bmatrix} 1 + i \\ 2 \\ -3\,i \end{bmatrix} \in \mathbb{C}^3, \qquad \underbrace{\overline{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}}_{3 \times 1} = \begin{bmatrix} 1 - i \\ 2 \\ 3\,i \end{bmatrix}. \]
The real entry \(2\) is its own conjugate. Every real vector is also a complex vector, since each real number \(a\) is the complex number \(a + 0\,i\), so \(\mathbb{R}^p\) is a subset of \(\mathbb{C}^p\).
Definition 3 (Complex conjugate of a complex vector or matrix) The complex conjugate of a complex vector or matrix takes the complex conjugate of each entry: \((\overline{\mathbf{A}})_{jk} \stackrel{\text{def}}{=}\overline{a_{jk}}\).
Definition 4 (Conjugate transpose, Hermitian adjoint, Hermitian conjugate, transjugate) The conjugate transpose [1] (also called the Hermitian adjoint, Hermitian conjugate, or transjugate) of an \(m \times n\) complex matrix \(\mathbf{A}\) is the \(n \times m\) matrix \(\mathbf{A}^{\mathsf{H}}\) obtained by transposing \(\mathbf{A}\) and then taking the complex conjugate of each entry:
\[(\mathbf{A}^{\mathsf{H}})_{jk} \stackrel{\text{def}}{=}\overline{a_{kj}} \quad \text{for } j \in \{1, \ldots, n\},\ k \in \{1, \ldots, m\}.\]
Remark 1 (Notation for the conjugate transpose). The operation is denoted variously across mathematics and physics: \(\mathbf{A}^*\) in pure mathematics (Axler [2], Definition 7.7, p. 231), \(\mathbf{A}^{\dagger}\) (“\(A\)-dagger”) in physics and quantum mechanics, and \(\mathbf{A}^{\mathsf{H}}\) in engineering, numerical linear algebra, and statistics. These notes write \(\mathbf{A}^{\mathsf{H}}\) because \(^*\) is reserved for optimal points (as in \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}^*\)) and \(^{\dagger}\) frequently denotes the Moore-Penrose pseudoinverse in the broader literature.
The Hermitian adjoint matrix should not be confused with the classical adjugate matrix (the transpose of the matrix of cofactors), which older texts occasionally referred to as the “adjoint.”
Example 2 (The conjugate transpose of a \(2 \times 3\) matrix) For
\[ \underbrace{\mathbf{A}}_{2 \times 3} = \begin{bmatrix} 1 + i & 2 & 0 \\ -i & 3 - 2\,i & 4 \end{bmatrix}, \]
transposing gives a \(3 \times 2\) matrix, and conjugating each of its entries gives
\[ \underbrace{\mathbf{A}^{\mathsf{H}}}_{3 \times 2} = \begin{bmatrix} 1 - i & i \\ 2 & 3 + 2\,i \\ 0 & 4 \end{bmatrix}. \]
For instance,
\[ \begin{aligned} (\mathbf{A}^{\mathsf{H}})_{12} &= \overline{a_{21}} \\ &= \overline{-i} \\ &= i. \end{aligned} \]
The transpose alone would have \(-i\) in that position, so for a matrix with an entry that is not real, \(\mathbf{A}^{\mathsf{H}}\) and \({\mathbf{A}}^{\top}\) differ. For a real matrix, conjugating changes nothing, so \(\mathbf{A}^{\mathsf{H}} = {\mathbf{A}}^{\top}\).
Remark 2 (Conjugate transpose of a complex vector). The conjugate transpose of a column vector \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} \in \mathbb{C}^p\) is the \(1 \times p\) row vector \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}} = [\overline{z_1},\; \ldots,\; \overline{z_p}]\).
Theorem 1 (Properties of the Hermitian adjoint) Let \(\mathbf{A}\) and \(\mathbf{B}\) be complex matrices of compatible dimensions, and let \(c \in \mathbb{C}\) be a scalar. The Hermitian adjoint (Definition 4) satisfies:
If \(\mathbf{A}\) is invertible, then \(\mathbf{A}^{\mathsf{H}}\) is also invertible, and
\[(\mathbf{A}^{\mathsf{H}})^{-1} = (\mathbf{A}^{-1})^{\mathsf{H}}.\]
Proof. Properties 1-3 follow directly from entrywise conjugation and transposition:
For involution:
\[ \begin{aligned} [(\mathbf{A}^{\mathsf{H}})^{\mathsf{H}}]_{jk} &= \overline{(\mathbf{A}^{\mathsf{H}})_{kj}} \\ &= \overline{\overline{a_{jk}}} \\ &= a_{jk}. \end{aligned} \]
For additivity:
\[ \begin{aligned} [(\mathbf{A} + \mathbf{B})^{\mathsf{H}}]_{jk} &= \overline{(\mathbf{A} + \mathbf{B})_{kj}} \\ &= \overline{a_{kj} + b_{kj}} \\ &= \overline{a_{kj}} + \overline{b_{kj}} \\ &= (\mathbf{A}^{\mathsf{H}})_{jk} + (\mathbf{B}^{\mathsf{H}})_{jk}. \end{aligned} \]
For conjugate-homogeneity:
\[ \begin{aligned} [(c\,\mathbf{A})^{\mathsf{H}}]_{jk} &= \overline{(c\,\mathbf{A})_{kj}} \\ &= \overline{c\,a_{kj}} \\ &= \overline{c}\,\overline{a_{kj}} \\ &= \overline{c}\,(\mathbf{A}^{\mathsf{H}})_{jk}. \end{aligned} \]
Unlike the real transpose where \({(c\,\mathbf{A})}^{\top} = c\,{\mathbf{A}}^{\top}\), scaling conjugates the scalar.
For property 4 (reversal of products), let \(\mathbf{A}\) be an \(m \times k\) matrix and \(\mathbf{B}\) a \(k \times n\) matrix. Entry \((i, j)\) of the product’s Hermitian adjoint (for \(i \in \{1, \ldots, n\}\) and \(j \in \{1, \ldots, m\}\)) is:
\[ \begin{aligned} [(\mathbf{A}\mathbf{B})^{\mathsf{H}}]_{ij} &= \overline{(\mathbf{A}\mathbf{B})_{ji}} && \text{(}\href{#def-conjugate-transpose}{\text{Definition~4}}\text{)} \\ &= \overline{\sum_{s=1}^{k} a_{js}\,b_{si}} && \text{(definition of matrix multiplication)} \\ &= \sum_{s=1}^{k} \overline{a_{js}}\,\overline{b_{si}} && \text{(conjugate of sums and products)} \\ &= \sum_{s=1}^{k} \overline{b_{si}}\,\overline{a_{js}} && \text{(complex multiplication is commutative)} \\ &= \sum_{s=1}^{k} (\mathbf{B}^{\mathsf{H}})_{is}\,(\mathbf{A}^{\mathsf{H}})_{sj} && \text{(}\href{#def-conjugate-transpose}{\text{Definition~4}}\text{)} \\ &= (\mathbf{B}^{\mathsf{H}}\mathbf{A}^{\mathsf{H}})_{ij} && \text{(definition of matrix multiplication)}. \end{aligned} \]
For the inverse, applying property 4 to \(\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}\) gives:
\[ \begin{aligned} (\mathbf{A}^{-1})^{\mathsf{H}}\mathbf{A}^{\mathsf{H}} &= (\mathbf{A}\mathbf{A}^{-1})^{\mathsf{H}} \\ &= \mathbf{I}^{\mathsf{H}} \\ &= \mathbf{I}, \end{aligned} \]
so \((\mathbf{A}^{\mathsf{H}})^{-1} = (\mathbf{A}^{-1})^{\mathsf{H}}\).
Theorem 2 (The adjoint property for inner products) Let \(\mathbf{A}\) be an \(m \times n\) complex matrix. Under the standard inner product on complex Euclidean space, \({\mathopen{}\left\langle \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} \stackrel{\text{def}}{=}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\),
\[{\mathopen{}\left\langle \mathbf{A}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} = {\mathopen{}\left\langle \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathbf{A}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} \quad \text{for all } \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}} \in \mathbb{C}^n \text{ and } \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} \in \mathbb{C}^m.\]
Proof. By the definition of the standard complex inner product and the product rule for the Hermitian adjoint (Theorem 1):
\[ \begin{aligned} {\mathopen{}\left\langle \mathbf{A}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} &= (\mathbf{A}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}})^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} && \text{(definition of standard inner product)} \\ &= (\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}^{\mathsf{H}}\mathbf{A}^{\mathsf{H}})\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} && \text{(}\href{#thm-hermitian-adjoint-properties}{\text{Theorem~1}}\text{, reversal of products)} \\ &= \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}^{\mathsf{H}}(\mathbf{A}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}) && \text{(associativity of matrix multiplication)} \\ &= {\mathopen{}\left\langle \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathbf{A}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} && \text{(definition of standard inner product)}. \end{aligned} \]
Remark 3 (Where the name “adjoint” comes from). This identity is the origin of the name adjoint: in functional analysis and operator theory, the adjoint of a linear map \(T: V \to W\) between inner product spaces is defined as the unique operator \(T^*: W \to V\) satisfying \({\mathopen{}\left\langle T\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} = {\mathopen{}\left\langle \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, T^*\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}}\) (Axler [2], Definition 7.1, p. 226). When \(V\) and \(W\) are finite-dimensional spaces equipped with their standard orthonormal bases, the matrix representing the adjoint operator \(T^*\) is precisely the Hermitian adjoint matrix \(\mathbf{A}^{\mathsf{H}}\) (Axler [2], Proposition 7.9, p. 231).
Theorem 3 (\(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}\) is a sum of squared absolute values) For every \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} \in \mathbb{C}^p\),
\[\underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}}}_{1 \times p}\,\underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}_{p \times 1} = \sum_{j=1}^p\mathopen{}\left|z_j\right|\mathclose{}^2,\]
which is a nonnegative real number, and is \(0\) only when \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} = \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}_{p \times 1}\).
Proof. The last step uses the identity for a complex number times its conjugate.
\[ \begin{aligned} \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} &= \sum_{j=1}^p\overline{z_j}\,z_j && \text{(}\href{#def-conjugate-transpose}{\text{Definition~4}}\text{, and the row-times-column product)} \\ &= \sum_{j=1}^pz_j\,\overline{z_j} && \text{(complex multiplication is commutative)} \\ &= \sum_{j=1}^p\mathopen{}\left|z_j\right|\mathclose{}^2 && \text{(a number times its conjugate)} \end{aligned} \]
By the definition of the absolute value, each \(\mathopen{}\left|z_j\right|\mathclose{}^2 = (\operatorname{Re} z_j)^2 + (\operatorname{Im} z_j)^2\) is a nonnegative real number, so their sum is too. A sum of nonnegative numbers is \(0\) only when every term is \(0\), and \(\mathopen{}\left|z_j\right|\mathclose{}^2 = 0\) only when
\[ \begin{aligned} \operatorname{Re} z_j &= \operatorname{Im} z_j \\ &= 0, \end{aligned} \]
that is, when \(z_j = 0\).
Example 3 (Computing \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}\)) Let \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} = {(1 + i, 2)}^{\top}\). By Theorem 3,
\[ \begin{aligned} \underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}}}_{1 \times 2}\,\underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}_{2 \times 1} &= \mathopen{}\left|1 + i\right|\mathclose{}^2 + \mathopen{}\left|2\right|\mathclose{}^2 && \text{(}\href{#thm-zhz-sum-squares}{\text{Theorem~3}}\text{)} \\ &= (1^2 + 1^2) + (2^2 + 0^2) && \text{(definition of the absolute value, squared)} \\ &= 2 + 4 && \text{(add inside each group)} \\ &= 6 && \text{(add)} \end{aligned} \]
Multiplying out directly gives the same number: \(\overline{2} \cdot 2 = 4\), and
\[ \begin{aligned} \overline{(1 + i)}\,(1 + i) &= (1 - i)(1 + i) && \text{(conjugate)} \\ &= 1 + i - i - i^2 && \text{(distribute)} \\ &= 1 - i^2 && \text{(cancel } i \text{)} \\ &= 2 && \text{(} i^2 = -1 \text{)} \end{aligned} \]
Example 4 (Why complex vectors use the conjugate transpose) For a real vector \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\), \({\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}}^{\top}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}= \sum_{j=1}^px_j^2\) is \(0\) only when \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\) is the zero vector. The same formula fails for complex vectors. Let \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} = {(1, i)}^{\top}\). Then
\[ \begin{aligned} \underbrace{{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}^{\top}}_{1 \times 2}\,\underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}_{2 \times 1} &= 1 \cdot 1 + i \cdot i && \text{(multiply entry by entry and add)} \\ &= 1 + i^2 && \text{(multiply)} \\ &= 0 && \text{(} i^2 = -1 \text{)} \end{aligned} \]
although \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}} \neq \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{0}}\vphantom{0}}_{2 \times 1}\). Using the conjugate transpose instead,
\[ \begin{aligned} \underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}^{\mathsf{H}}}_{1 \times 2}\,\underbrace{\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{z}}\vphantom{z}}}_{2 \times 1} &= \overline{1} \cdot 1 + \overline{i} \cdot i && \text{(}\href{#def-conjugate-transpose}{\text{Definition~4}}\text{)} \\ &= 1 + (-i) \cdot i && \text{(conjugate each entry)} \\ &= 1 - i^2 && \text{(multiply)} \\ &= 2 && \text{(} i^2 = -1 \text{)} \end{aligned} \]
which is positive, as Theorem 3 guarantees.
Definition 5 (Hermitian matrix, self-adjoint matrix) A square complex matrix \(\mathbf{A} \in \mathbb{C}^{n \times n}\) is Hermitian [3] (or self-adjoint) if it equals its conjugate transpose (Definition 4):
\[\mathbf{A}^{\mathsf{H}} = \mathbf{A}.\]
Remark 4 (Why “self-adjoint”). The term self-adjoint comes directly from Theorem 2: when \(\mathbf{A}\) is Hermitian, \({\mathopen{}\left\langle \mathbf{A}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}} = {\mathopen{}\left\langle \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathbf{A}\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}}\right\rangle\mathclose{}}\) for all \(\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{u}}\vphantom{u}}, \mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{v}}\vphantom{v}} \in \mathbb{C}^n\). The linear operator moves across the inner product without change.
Example 5 (A Hermitian matrix) \[ \underbrace{\mathbf{A}}_{2 \times 2} = \begin{bmatrix} 2 & 1 - i \\ 1 + i & 3 \end{bmatrix} \]
is Hermitian: transposing swaps the off-diagonal entries, and conjugating turns \(1 + i\) back into \(1 - i\) and \(1 - i\) back into \(1 + i\), so \(\mathbf{A}^{\mathsf{H}} = \mathbf{A}\). The diagonal entries of a Hermitian matrix must be real, because \(a_{jj} = \overline{a_{jj}}\), and only a real number is its own conjugate: if \(a + b\,i = a - b\,i\), then the imaginary parts give \(b = -b\), so \(b = 0\).
Every real symmetric matrix is Hermitian, because for a real matrix \(\mathbf{A}^{\mathsf{H}} = {\mathbf{A}}^{\top}\).
Example 6 (A symmetric complex matrix that is not Hermitian) \[ \underbrace{\mathbf{B}}_{2 \times 2} = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \]
is symmetric, \({\mathbf{B}}^{\top} = \mathbf{B}\), but it is not Hermitian:
\[ \underbrace{\mathbf{B}^{\mathsf{H}}}_{2 \times 2} = \begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix} \neq \mathbf{B}. \]
Definition 6 (Skew-Hermitian matrix, anti-Hermitian matrix) A square complex matrix \(\mathbf{A} \in \mathbb{C}^{n \times n}\) is skew-Hermitian [4] (or anti-Hermitian) if it equals the negative of its conjugate transpose (Definition 4):
\[\mathbf{A}^{\mathsf{H}} = -\mathbf{A}.\]
Remark 5 (The diagonal of a skew-Hermitian matrix). Every entry of a skew-Hermitian matrix satisfies \(a_{jk} = -\overline{a_{kj}}\). In particular, its diagonal entries satisfy \(a_{jj} = -\overline{a_{jj}}\), which means
\[ \begin{aligned} a_{jj} + \overline{a_{jj}} &= 2\operatorname{Re}(a_{jj}) \\ &= 0. \end{aligned} \]
Therefore, every diagonal entry of a skew-Hermitian matrix must be purely imaginary (or zero). For example, \(a_{jj} = 3\,i\) works, since
\[ \begin{aligned} -\overline{3\,i} &= -(-3\,i) \\ &= 3\,i, \end{aligned} \]
but \(a_{jj} = 3\) does not, since \(-\overline{3} = -3\).
Example 7 (A skew-Hermitian matrix) The matrix
\[ \mathbf{A} = \begin{bmatrix} 2\,i & 1 + i \\ -1 + i & 0 \end{bmatrix} \]
is skew-Hermitian: transposing and conjugating gives
\[ \begin{aligned} \mathbf{A}^{\mathsf{H}} &= \begin{bmatrix} \overline{2\,i} & \overline{-1 + i} \\ \overline{1 + i} & \overline{0} \end{bmatrix} \\ &= \begin{bmatrix} -2\,i & -1 - i \\ 1 - i & 0 \end{bmatrix} \\ &= -\mathbf{A}. \end{aligned} \]
Its diagonal entries \(2\,i\) and \(0\) are purely imaginary.
Theorem 4 (Hermitian and skew-Hermitian decomposition) Every square complex matrix \(\mathbf{A} \in \mathbb{C}^{n \times n}\) can be uniquely expressed as the sum of a Hermitian matrix \(\mathbf{H}\) (Definition 5) and a skew-Hermitian matrix \(\mathbf{S}\) (Definition 6):
\[\mathbf{A} = \mathbf{H} + \mathbf{S},\]
where
\[\mathbf{H} = \frac{\mathbf{A} + \mathbf{A}^{\mathsf{H}}}{2} \quad \text{and} \quad \mathbf{S} = \frac{\mathbf{A} - \mathbf{A}^{\mathsf{H}}}{2}.\]
Proof. Existence: Define \(\mathbf{H} = \frac{1}{2}(\mathbf{A} + \mathbf{A}^{\mathsf{H}})\) and \(\mathbf{S} = \frac{1}{2}(\mathbf{A} - \mathbf{A}^{\mathsf{H}})\). Their sum is:
\[ \begin{aligned} \mathbf{H} + \mathbf{S} &= \frac{\mathbf{A} + \mathbf{A}^{\mathsf{H}}}{2} + \frac{\mathbf{A} - \mathbf{A}^{\mathsf{H}}}{2} \\ &= \frac{2\mathbf{A}}{2} \\ &= \mathbf{A}. \end{aligned} \]
Applying the properties of the Hermitian adjoint (Theorem 1):
\[ \begin{aligned} \mathbf{H}^{\mathsf{H}} &= \mathopen{}\left(\frac{\mathbf{A} + \mathbf{A}^{\mathsf{H}}}{2}\right)\mathclose{}^{\mathsf{H}} \\ &= \frac{\mathbf{A}^{\mathsf{H}} + (\mathbf{A}^{\mathsf{H}})^{\mathsf{H}}}{2} \\ &= \frac{\mathbf{A}^{\mathsf{H}} + \mathbf{A}}{2} \\ &= \mathbf{H}, \\ \mathbf{S}^{\mathsf{H}} &= \mathopen{}\left(\frac{\mathbf{A} - \mathbf{A}^{\mathsf{H}}}{2}\right)\mathclose{}^{\mathsf{H}} \\ &= \frac{\mathbf{A}^{\mathsf{H}} - (\mathbf{A}^{\mathsf{H}})^{\mathsf{H}}}{2} \\ &= \frac{\mathbf{A}^{\mathsf{H}} - \mathbf{A}}{2} \\ &= -\mathbf{S}. \end{aligned} \]
Thus \(\mathbf{H}\) is Hermitian and \(\mathbf{S}\) is skew-Hermitian.
Uniqueness: Suppose \(\mathbf{A} = \mathbf{H}' + \mathbf{S}'\) with \((\mathbf{H}')^{\mathsf{H}} = \mathbf{H}'\) and \((\mathbf{S}')^{\mathsf{H}} = -\mathbf{S}'\). Taking the Hermitian adjoint of both sides yields:
\[ \begin{aligned} \mathbf{A}^{\mathsf{H}} &= (\mathbf{H}' + \mathbf{S}')^{\mathsf{H}} \\ &= (\mathbf{H}')^{\mathsf{H}} + (\mathbf{S}')^{\mathsf{H}} \\ &= \mathbf{H}' - \mathbf{S}'. \end{aligned} \]
Adding \(\mathbf{A} = \mathbf{H}' + \mathbf{S}'\) and \(\mathbf{A}^{\mathsf{H}} = \mathbf{H}' - \mathbf{S}'\) gives \(\mathbf{A} + \mathbf{A}^{\mathsf{H}} = 2\mathbf{H}'\), so
\[ \begin{aligned} \mathbf{H}' &= \frac{1}{2}(\mathbf{A} + \mathbf{A}^{\mathsf{H}}) \\ &= \mathbf{H}. \end{aligned} \]
Subtracting the two equations gives \(\mathbf{A} - \mathbf{A}^{\mathsf{H}} = 2\mathbf{S}'\), so
\[ \begin{aligned} \mathbf{S}' &= \frac{1}{2}(\mathbf{A} - \mathbf{A}^{\mathsf{H}}) \\ &= \mathbf{S}. \end{aligned} \]