Conjugate transpose of a matrix

Since we now have the definition of the complex conjugate of a number, I'd like to quickly go over the conjugate transpose of a matrix as we will use this later in the book. The conjugate transpose is exactly as it sounds. It combines the notions of complex conjugates and the transposition of a matrix into one operation. If you remember from Chapter 2, The Matrix, we defined the transpose to be:

This is where we essentially convert the rows into columns and the columns into rows.

The conjugate of a matrix is just the conjugation of every entry:

For example, if the matrix M equals

,

then M* equals

.

So here is the big payoff. The conjugate transpose of a matrix A is defined to be:

The cross symbol at the top right of A is pronounced "dagger," and therefore when you hear "A dagger," the conjugate transpose of A is being referred to.

A quick example should get this all sorted. Let's use...