Conjugate Transpose · aiwiki.page

The conjugate transpose of a matrix is obtained by interchanging its rows and columns and taking the complex conjugate of every entry. It combines transposition with conjugation and is commonly denoted by $A^*$ , $A^H$ , or $A^\dagger$ . In complex linear algebra, it plays the role that ordinary transposition plays for real matrices, particularly in inner products and the definition of adjoint transformations. (reference.wolfram.com)

Definition and notation

For an $m\times n$ matrix $A=(a_{ij})$ with complex entries, its conjugate transpose is the $n\times m$ matrix defined by

(A^*)_{ij}=\overline{a_{ji}}.

Here complex conjugation sends $a+bi$ to $a-bi$ , where $a,b$ are real and $i^2=-1$ . Conjugation and transposition commute, so

A^*=\overline{A}^{\,T}=\overline{A^T}.

If all entries are real numbers, conjugation leaves them unchanged, giving $A^*=A^T$ . The operation applies to rectangular as well as square matrices. (reference.wolfram.com)

For example, applying the definition gives

A= \begin{pmatrix} 1+i&2&-i\\ 3i&4-i&5 \end{pmatrix}, \qquad A^*= \begin{pmatrix} 1-i&-3i\\ 2&4+i\\ i&5 \end{pmatrix}.

Thus each column of $A^*$ is the conjugate of the corresponding row of $A$ .

Notation requires attention: an asterisk sometimes denotes entrywise conjugation rather than conjugate transposition. The dagger notation is another established convention, while $A^H$ explicitly indicates the Hermitian transpose. (reference.wolfram.com)

Algebraic properties

For matrices of compatible sizes and a complex scalar $\alpha$ ,

(A^*)^*=A,\qquad (A+B)^*=A^*+B^*,\qquad (\alpha A)^*=\overline{\alpha}A^*.

The first identity makes the operation an involution: applying it twice returns the original matrix. The scalar identity shows that it is conjugate-linear, rather than complex-linear, on the vector space of matrices of a fixed size. (math.ucdavis.edu)

For matrix multiplication, the order of factors reverses:

(AB)^*=B^*A^*.

This reversal is essential; in general, $A^*B^*$ is neither the correct result nor necessarily dimensionally meaningful. If $A$ has an inverse, then

(A^{-1})^*=(A^*)^{-1}.

These rules allow conjugate transposes of complicated expressions to be calculated without expanding every entry. (fab.cba.mit.edu)

Inner products and adjoints

For column vectors $x,y\in\mathbb C^n$ , the standard complex inner product is

\langle x,y\rangle=x^*y =\sum_{j=1}^n\overline{x_j}y_j.

This convention is conjugate-linear in its first argument and linear in its second. In particular,

\langle x,x\rangle=\sum_j|x_j|^2,

which is positive for every nonzero vector. It defines the Euclidean norm by $\|x\|_2=\sqrt{x^*x}$ . Unlike the real case, using $x^Tx$ would not provide a positive squared length: for $x=(1,i)^T$ , it equals zero although $x\ne0$ . (web.mit.edu)

For a matrix representing a linear map $A:\mathbb C^n\to\mathbb C^m$ , the product rule yields

\langle Ax,y\rangle=\langle x,A^*y\rangle.

This identity characterizes its adjoint under the standard inner products. Thus conjugate transposition transfers a transformation from one argument of an inner product to the other. The identification is tied to orthonormal coordinates; an arbitrary basis does not generally represent the adjoint by simply conjugate-transposing the original matrix. (reference.wolfram.com)

Important classes of matrices

A square Hermitian matrix satisfies $A^*=A$ . Its diagonal entries are real, and entries reflected across the diagonal are complex conjugates. Hermitian matrices have real eigenvalues and an orthonormal basis of eigenvectors. This is a finite-dimensional form of the spectral theorem, connecting conjugate transposition with diagonalization. (web.mit.edu)

A unitary matrix satisfies

U^*U=UU^*=I,

where $I$ is the identity matrix. Equivalently, $U^{-1}=U^*$ . Its columns are orthonormal, and it preserves inner products and vector norms. When its entries are real, the condition reduces to that defining an orthogonal matrix. (math.ucdavis.edu)

For any rectangular matrix $A$ , the matrix $A^*A$ is Hermitian. Applying the inner-product identity gives

x^*A^*Ax=\|Ax\|_2^2\ge0.

Consequently, it is positive semidefinite. If the columns of $A$ are linearly independent, the expression is strictly positive for nonzero $x$ , making $A^*A$ a positive-definite matrix. Its entries are the pairwise inner products of the columns, so it is their Gram matrix. These conclusions follow directly from the preceding identities. (web.mit.edu)

Role in matrix decompositions

In numerical linear algebra, conjugate transposition appears in the complex singular value decomposition:

A=U\Sigma V^*.

Here $U$ and $V$ are unitary, while $\Sigma$ is rectangular diagonal with nonnegative real entries, the singular values. The columns of $U$ and $V$ are respectively the left and right singular vectors. LAPACK's complex SVD routine returns $V^*$ , rather than $V$ , reflecting the form used in reconstruction. (netlib.org)