Transpose & Conjugate Transpose

Definitions

The transpose of a matrix $A$ is $A\trps$ with entries $$A\trps_{ij} = A_{ji}$$ and the conjugate transpose of $A$ is $A^*$ (or $A^\textrm{H}$) with entries $$A^*_{ij} = \baa{A_{ji}}$$

We have $A^* = \baa{A\trps}$. And when $A$ is real, $A^* = A\trps$.

Special Matrices

Let $A$ be a square matrix. The following properties have fancy names.

  • Symmetric $$A\trps = A$$
  • Hermitian $$A\trps = A^*$$
  • Skew-Symmetric $$A\trps = -A$$
  • Skew-Hermitian $$A\trps = -A^*$$
  • Orthogonal (although the columns are orthonormal) $$A\trps = A^{-1}$$
  • Unitary $$A^* = A^{-1}$$
  • Normal ($\supset$ Unitary) $$A^*A = AA^*$$

Reference

Exported: 2016-07-13T01:31:22.401744