Complex Matrices

The matrices with the elements coming from the complex number system are known as Complex Matrices.

Inner product of Complex Vectors

Recall, in \(\R^n,\) \(x.y=x^Ty.\)

But in \(\mathbb{C},\) we cannot use the same definition.

So we define the inner product as \(x.y=\bar{x}^Ty\)

Warning

\(\bar{x}^Ty\neq\bar{y}^Tx\)

Recall, in \(\R^n,\) \(||x||^2=x^Tx.\)

But in \(\mathbb{C}^n,\) we cannot use the same definition.

So we define the length as \(||x||^2=\bar{x}^Tx\)

For a complex matrix \(A\), the conjugate transpose \(A^*\) is defined as