Linear Algebra with Applications, 10th Edition

6.4 Hermitian Matrices

Let ${ℂ}^n$ denote the vector space of all n-tuples of complex numbers. The set ℂ of all complex numbers will be taken as our field of scalars. We have already seen that a matrix A with real entries may have complex eigenvalues and eigenvectors. In this section, we study matrices with complex entries and look at the complex analogues of symmetric and orthogonal matrices.

Complex Inner Products

If $\alpha =a+bi$ is a complex scalar, the length of $\alpha$ is given by

$$\left|\alpha \right|=\sqrt{\overline{\alpha }\alpha }=\sqrt{a^2+b^2}$$

The length of a vector $\mathbf{\text{z}}=(z_1,z_2,\dots ,z_n{)}^T$ in ${ℂ}^n$ is given by

$$\begin{array}{cc}\hfill \Vert \mathbf{\text{z}}\Vert & =(|z_1{|}^2+|z_2{|}^2+\dots +|z_n{|}^2{)}^{1/2}\hfill \\ ={({\overline{z}}_1{z}_1+{\overline{z}}_2{z}_2+\dots +{\overline{z}}_n{z}_n)}^{1/2}\hfill \\ =({\overline{\mathbf{\text{z}}}}^T\mathbf{\text{z}}{)}^{1/2}\hfill \end{array}$$

As a notational convenience, we write ${\mathbf{\text{z}}}^H$ for the transpose of $\overline{\mathbf{\text{z}}}$. Thus,

$$\begin{array}{ccc}{\overline{\mathbf{\text{z}}}}^T={\mathbf{\text{z}}}^H& \text{and}& \Vert \mathbf{\text{z}}\Vert =({\mathbf{\text{z}}}^H\mathbf{\text{z}}{)}^{1/2}\end{array}$$

Definition

Let V be a vector space over ...