Conjugate transpose

Suppose that a matrix $M$ contains complex numbers. Then the conjugate transpose of $M$ is a matrix $M^{H}$ containing the complex conjugates of the matrix transposition of $M$.

Conjugate transpose is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

$(M^{H})_{ji}={\overline {M_{ij}}}$

This means that row $j$, column $i$ of the conjugate transpose equals the complex conjugate of row $i$, column $j$ of the original matrix.

In the next list, $M$ must also be a square matrix.

• A Hermitian matrix equals its own conjugate transpose: $M^{H}=M$.
• A normal matrix is commutative in multiplication with its conjugate transpose: $M^{H}M=MM^{H}$.
• A unitary matrix has its inverse equal to its conjugate transpose: $M^{H}=M^{-1}$. This is true iff $M^{H}M=I_{n}$ and iff $MM^{H}=I_{n}$, where $I_{n}$ is the identity matrix.

Given some matrix of complex numbers, find its conjugate tranpose. Also determine if it is a Hermitian matrix, normal matrix, or a unitary matrix.

• MathWorld: conjugate transpose, Hermitian matrix, normal matrix, unitary matrix