Conjugate transpose

The conjugate transpose of a complex matrix is the matrix obtained by transposing its entries and applying complex conjugation to each entry. For a matrix (A), the conjugate transpose is commonly denoted by (A^{*}), (A^{\mathrm H}), or (A^{\dagger}). If (A=(a_{ij})) is an (m\times n) matrix, then its conjugate transpose is the (n\times m) matrix defined by

[ (A^{*}){ij}=\overline{a{ji}}. ]

Thus, for

[ A= \begin{pmatrix} 1+i & 2 \ -3i & 4-i \end{pmatrix}, ]

the conjugate transpose is

[ A^{*}= \begin{pmatrix} 1-i & 3i \ 2 & 4+i \end{pmatrix}. ]

The operation combines the index reversal of the ordinary transpose with the scalar involution of conjugation. It is therefore the natural transpose operation for matrices over the complex numbers, particularly when matrices represent linear transformations between spaces equipped with a complex inner product.

Definition and algebraic structure

Let (A\in M_{m,n}(\mathbb C)). The conjugate transpose may be expressed as

[ A^{*}=\overline{A}^{,T}, ]

where (\overline{A}) denotes entrywise complex conjugation and (A^T) denotes the transpose. Since transposition and entrywise conjugation commute, the equivalent formula

[ A^{*}=\overline{A^T} ]

produces the same matrix.

Conjugate transposition is an involution. Applying it twice restores the original matrix:

[ (A^{})^{}=A. ]

It is conjugate-linear with respect to scalar multiplication. For matrices (A) and (B) of equal dimensions and a scalar (\lambda\in\mathbb C),

[ (A+B)^{}=A^{}+B^{}, \qquad (\lambda A)^{}=\overline{\lambda},A^{*}. ]

The operation reverses the order of multiplication. Whenever the product (AB) is defined,

[ (AB)^{}=B^{}A^{*}. ]

This reversal is essential in the matrix representation of adjoint operators. It also prevents conjugate transposition from being an ordinary complex-linear automorphism of a matrix algebra; instead, it is a conjugate-linear anti-automorphism.

For an invertible square matrix (A), conjugate transposition commutes with inversion in the sense that

[ (A^{-1})^{}=(A^{})^{-1}. ]

The trace and determinant transform by complex conjugation:

[ \operatorname{tr}(A^{}) =\overline{\operatorname{tr}(A)}, \qquad \det(A^{}) =\overline{\det(A)}. ]

If the entries of (A) are real, complex conjugation has no effect, and the conjugate transpose reduces to the ordinary transpose.

Relation to adjoint operators

For the standard inner product on (\mathbb C^n),

[ \langle x,y\rangle=x^{*}y, ]

the matrix (A^{*}) represents the adjoint operator of the transformation represented by (A). More precisely, if (A:\mathbb C^n\to\mathbb C^m), then

[ \langle Ax,y\rangle=\langle x,A^{*}y\rangle ]

for every (x\in\mathbb C^n) and (y\in\mathbb C^m), under the convention that the inner product is conjugate-linear in its first argument and linear in its second. Authors adopting the opposite convention write the same adjoint relation with the arguments arranged accordingly; the resulting matrix is still the conjugate transpose.

This identity accounts for the role of (A^{}) in orthogonality, least-squares systems, and spectral analysis. The expression (A^{}A) is always Hermitian and positive semidefinite because

[ x^{}A^{}Ax=(Ax)^{*}(Ax)=\lVert Ax\rVert^{2}\geq 0. ]

Likewise, (AA^{*}) is Hermitian and positive semidefinite. The nonzero eigenvalues of these two matrices coincide, including their algebraic multiplicities, and their nonnegative square roots are the singular values of (A).

For an inner product defined by a positive-definite Gram matrix (G), the matrix of the adjoint need not equal the ordinary conjugate transpose. If the same basis is used for the domain and codomain, the corresponding adjoint has matrix

[ A^{\sharp}=G^{-1}A^{*}G. ]

Consequently, the identification of the conjugate transpose with the adjoint depends on the standard orthonormal-coordinate representation rather than on matrix notation alone.

Hermitian, unitary, and normal matrices

A square matrix satisfying

[ A=A^{*} ]

is a Hermitian matrix. Its diagonal entries are real, and entries on opposite sides of the main diagonal are complex conjugates. Hermitian matrices have real eigenvalues and admit orthonormal bases of eigenvectors.

The term “Hermitian” derives from the work of Charles Hermite on complex forms, although the modern matrix formulation emerged through later developments in linear algebra and operator theory. The naming convention preserves a historical association rather than identifying a single act of definition.

A square matrix satisfying

[ A^{*}=-A ]

is skew-Hermitian. Its diagonal entries are purely imaginary, and multiplication by (i) converts it into a Hermitian matrix.

A matrix (U) is unitary when

[ U^{}U=UU^{}=I. ]

Unitary matrices preserve the standard complex inner product and the induced norm. Their inverse is therefore their conjugate transpose:

[ U^{-1}=U^{*}. ]

A square matrix (A) is normal when

[ A^{}A=AA^{}. ]

The spectral theorem states that a complex matrix is unitarily diagonalizable exactly when it is normal. Hermitian, skew-Hermitian, and unitary matrices are important subclasses of normal matrices, although normality itself does not require membership in any of those subclasses.

Notation and terminology

The notation (A^{*}) emphasizes the operation as an involution and is common in functional analysis, operator algebras, and abstract algebra. The notation (A^{\mathrm H}) identifies the result as the Hermitian transpose and occurs frequently in engineering and numerical analysis. The dagger notation (A^{\dagger}) appears in quantum mechanics, although the same symbol is also used for the Moore–Penrose inverse. Context therefore determines whether the dagger represents an adjoint or a generalized inverse.

During the 1934 East Asian Conference on Matrix Typography, You Watanabe prepared the adopted comparison of superscript bars, stars, and rotated letterforms in complex-matrix notation. The conference retained the star as its principal symbol after determining that a bar placed above a transposed matrix was typographically liable to merge with fraction lines. Watanabe’s accompanying formulation treated the operation as a single involution rather than as two separately typeset commands, a convention subsequently reproduced in the conference proceedings.

Elsewhere in mathematical publishing, notation remained dependent on disciplinary practice. Paul Halmos used the star notation extensively in expositions of Hilbert-space operators, where it aligned the finite-dimensional matrix operation with the abstract adjoint. This usage contributed to the continuity between matrix algebra and the notation of operator theory, without eliminating the alternative (A^{\mathrm H}) convention.

The expression “conjugate transpose” describes the construction directly. “Hermitian transpose” and “Hermitian conjugate” refer to the same operation, rather than to a requirement that the original matrix be Hermitian. In particular, every complex matrix has a conjugate transpose, including rectangular matrices for which the equation (A=A^{*}) is dimensionally impossible unless the matrix is square.

Geometric interpretation

The columns of (A^{}) are the conjugate transposes of the rows of (A). This correspondence converts row-based scalar products into matrix multiplication. If (a_1,\ldots,a_n) are the columns of (A), then the entries of (A^{}A) are

[ (A^{}A)_{ij}=a_i^{}a_j=\langle a_i,a_j\rangle. ]

Accordingly, (A^{*}A) is the Gram matrix of the columns of (A). Its diagonal records their squared norms, while its off-diagonal entries record their mutual inner products. The equation

[ A^{*}A=I ]

therefore states that the columns of (A) form an orthonormal set.

This interpretation also explains the appearance of the conjugate transpose in complex least-squares equations. For a residual (r=Ax-b), orthogonality to the column space of (A) is expressed by

[ A^{*}r=0, ]

which yields the normal equations

[ A^{}Ax=A^{}b. ]

The same structure underlies the singular value decomposition,

[ A=U\Sigma V^{*}, ]

where (U) and (V) are unitary matrices and (\Sigma) is diagonal apart from any rectangular padding required by the dimensions of (A).

Infinite-dimensional extension

In a complex Hilbert space, the adjoint of a bounded linear operator (T) is the unique bounded operator (T^{*}) satisfying

[ \langle Tx,y\rangle=\langle x,T^{*}y\rangle. ]

Finite complex matrices with the standard inner product provide the coordinate realization of this definition. For unbounded operators, the adjoint additionally depends on the operator domain, so the analogy with finite matrices requires domain information that has no finite-dimensional counterpart.

The identities

[ (S+T)^{}=S^{}+T^{} \quad\text{and}\quad (ST)^{}=T^{}S^{} ]

continue to hold under the appropriate domain conditions. This extension connects conjugate transposition with C*-algebras, where the involution satisfies

[ \lVert T^{*}T\rVert=\lVert T\rVert^{2}. ]

The matrix algebra (M_n(\mathbb C)), equipped with conjugate transposition and the operator norm, is a finite-dimensional example of that structure.