Conjugation And Conjugate Transpose

Two Operations on a Complex Matrix

For a real matrix, transposing is the familiar operation that switches rows and columns. With complex entries, there is another operation to keep track of: complex conjugation changes the sign of each imaginary part. Combining these operations gives the conjugate transpose. Careful notation matters because transposing alone does not conjugate entries, and conjugating a product does not reverse its factors in the same way as taking a conjugate transpose.

For a complex number \(z=a+bi\), where \(a,b\in\mathbb{R}\) and \(i^2=-1\), its conjugate is \(\overline{z}=a-bi\). The scalar identities used below follow by multiplying these expressions: \(\overline{z+w}=\overline z+\overline w\), \(\overline{zw}=\overline z\,\overline w\), and \(\overline{cz}=\overline c\,\overline z\). For example, if \(z=a+bi\) and \(w=d+ei\), then \(zw=(ad-be)+(ae+bd)i\), whose conjugate is \((ad-be)-(ae+bd)i=(a-bi)(d-ei)\).

Thus, taking a conjugate changes entries but not the matrix shape; transposing changes the shape but not the entries; and taking a conjugate transpose does both. For a real matrix, every entry is unchanged by conjugation, so \(\mathbf{A}^{*}=\mathbf{A}^{\top}\). For a complex matrix, that equality need not hold.

Identities for Sums, Scalars, and Transposes

The next identities apply to complex matrices of the indicated sizes. In each sum, the matrices must have the same size; in each product, the number of columns of the first factor must equal the number of rows of the second. These conditions ensure that both sides of an identity are defined and have the same size.

Proof. Write \(\mathbf{A}=[a_{ij}]\) and \(\mathbf{B}=[b_{ij}]\). The \((i,j)\) entry of \(\overline{\mathbf{A}+\mathbf{B}}\) is \(\overline{a_{ij}+b_{ij}}=\overline{a_{ij}}+\overline{b_{ij}}\), which is the \((i,j)\) entry of \(\overline{\mathbf{A}}+\overline{\mathbf{B}}\). The \((i,j)\) entry of \(\overline{c\mathbf{A}}\) is \(\overline{ca_{ij}}=\overline c\,\overline{a_{ij}}\), giving the scalar identity. By definition, the \((i,j)\) entry of \(\mathbf{A}^{*}\) is \(\overline{a_{ji}}\), which is also the \((i,j)\) entry of both \((\overline{\mathbf{A}})^{\top}\) and \(\overline{\mathbf{A}^{\top}}\). The \((i,j)\) entry of \((\mathbf{A}^{\top})^{*}\) is \(\overline{(\mathbf{A}^{\top})_{ji}}=\overline{a_{ij}}\), giving \((\mathbf{A}^{\top})^{*}=\overline{\mathbf{A}}\). Finally, the \((i,j)\) entry of \((\mathbf{A}^{*})^{*}\) is \(\overline{(\mathbf{A}^{*})_{ji}}=\overline{\overline{a_{ij}}}=a_{ij}\), so \((\mathbf{A}^{*})^{*}=\mathbf{A}\). Applying the definition to a sum or scalar multiple, together with the entrywise identities already proved, gives the last two identities. \(\square\)

The same entry-by-entry method proves the product rules. Their proof is worth examining because it explains the difference between conjugating a product and taking its conjugate transpose.

Proof. By the definition of matrix multiplication, the \((i,j)\) entry of \(\mathbf{A}\mathbf{B}\) is \(\sum_{k=1}^{n}a_{ik}b_{kj}\). Its conjugate is \(\sum_{k=1}^{n}\overline{a_{ik}}\,\overline{b_{kj}}\), which is the \((i,j)\) entry of \(\overline{\mathbf{A}}\,\overline{\mathbf{B}}\). This proves the first identity. For the second, the \((i,j)\) entry of \((\mathbf{A}\mathbf{B})^{*}\) is the conjugate of the \((j,i)\) entry of \(\mathbf{A}\mathbf{B}\), namely \[ \overline{\sum_{k=1}^{n}a_{jk}b_{ki}} =\sum_{k=1}^{n}\overline{b_{ki}}\,\overline{a_{jk}}. \] The \((i,j)\) entry of \(\mathbf{B}^{*}\mathbf{A}^{*}\) is \(\sum_{k=1}^{n}(\mathbf{B}^{*})_{ik}(\mathbf{A}^{*})_{kj} =\sum_{k=1}^{n}\overline{b_{ki}}\,\overline{a_{jk}}\). The entries agree for every \(i,j\). The product \(\mathbf{B}^{*}\mathbf{A}^{*}\) has size \(p\times m\), the same size as \((\mathbf{A}\mathbf{B})^{*}\), so the identity is defined with matching dimensions. \(\square\)

Checking the Product Rules by Calculation

Rectangular Matrices and the Role of Dimensions

The identities do not require square matrices. Dimensions are particularly helpful for detecting mistakes: if \(\mathbf{A}\) is \(m\times n\), then \(\mathbf{A}^{*}\) is \(n\times m\). If \(\mathbf{A}\) is \(m\times n\) and \(\mathbf{B}\) is \(n\times p\), then \(\mathbf{B}^{*}\mathbf{A}^{*}\) has size \(p\times m\), just like \((\mathbf{A}\mathbf{B})^{*}\). By contrast, \(\mathbf{A}^{*}\mathbf{B}^{*}\) may not even be defined.

Common Mistake: Forgetting to Reverse the Product

A common incorrect calculation is \((\mathbf{A}\mathbf{B})^{*}=\mathbf{A}^{*}\mathbf{B}^{*}\). The conjugate transpose does conjugate and transpose, but transposing a product reverses the order of its factors. The correct identity is \((\mathbf{A}\mathbf{B})^{*}=\mathbf{B}^{*}\mathbf{A}^{*}\), as the entrywise proof established.

For the matrices in the previous product example, the incorrect order gives a concrete contradiction:

But the correct result is \((\mathbf{A}\mathbf{B})^{*}=\begin{bmatrix}1-3i&1\\1+i&-1\end{bmatrix}\). These matrices are unequal, so the claimed identity fails. The correction is not to change the conjugation of entries, but to reverse the factor order: compute \(\mathbf{B}^{*}\mathbf{A}^{*}\).

Another useful distinction is between \(\mathbf{A}^{\top}\) and \(\mathbf{A}^{*}\). For the complex matrix \(\mathbf{C}\) above, the entry \(i\) stays \(i\) in \(\mathbf{C}^{\top}\), but becomes \(-i\) in \(\mathbf{C}^{*}\). Using a transpose where a conjugate transpose is required loses that sign change; conjugating without transposing loses the required change in shape.

Why These Identities Are Useful

Conjugation and conjugate transpose let complex matrix calculations be organized using familiar entrywise operations. The product identities are especially important: \(\overline{\mathbf{A}\mathbf{B}}=\overline{\mathbf{A}}\,\overline{\mathbf{B}}\) preserves the order, but \((\mathbf{A}\mathbf{B})^{*}=\mathbf{B}^{*}\mathbf{A}^{*}\) reverses it. The distinction comes from the transpose, not from scalar conjugation. Keeping the definitions and dimensions visible provides a reliable check before applying these operations in more involved calculations.

When checking an identity, a useful routine is to record the dimensions first, then compute entries using the definitions. For an equality involving a product, compare a typical \((i,j)\) entry on each side; for a conjugate transpose, remember that the entry comes from the transposed position and is then conjugated. This method works for square and rectangular complex matrices alike.