Matrices with Real or Complex Entries

Abstract

A square matrix M∈M_n(ℝ) is said to be normal if M and M^T commute: M^TM = MM^T . The real symmetric, skew-symmetric, and orthogonal matrices are normal.

In considering matrices with complex entries, a useful operation is complex conjugation$z \mapsto \bar z$. One denotes by $\bar M$ the matrix obtained from M by conjugating the entries. We then define the Hermitian adjoint matrix1 M^* by $M^* : = \left( {\bar M} \right)^T = \overline {M^T .}$

One therefore has $m_{ij}^* = \overline {m_{ji} }$ and det $M^* = \overline {\det M}$. The map M ↦ M* is an anti-isomorphism, which means that it is antilinear (meaning that $\left( {\lambda M} \right)^* = \bar \lambda M^* )$ and satisfies, moreover, the product formula $\left( {MN} \right)^* = N^* M^* .$.

When a square matrix M∈M_n(ℂ) is invertible, then $\left( {M^* } \right)^{ - 1} = \left( {M^{ - 1} } \right)^*$. This matrix is sometimes denoted by M^-*.