Matrices

A matrix is defined as a rectangular array of numbers, e.g.,

$$\mathbf{A}= \left[\begin{array}{cc} a & b \\ [2pt] c & d \end{array}\right]$$

which is a $2\times2$ (``two by two'') matrix. A general matrix may be $M\times N$, where $M$ is the number of rows, and $N$ is the number of columns of the matrix. For example, the general $3\times 2$ matrix is

$$\left[\begin{array}{cc} a & b \\ c & d \\ e & f \end{array}\right].$$

Either square brackets or large parentheses may be used to delimit the matrix. The $(i,j)$th element^H.1 of a matrix $\mathbf{A}$ may be denoted by $\mathbf{A}[i,j]$, $\mathbf{A}(i,j)$, or $\mathbf{A}_{ij}$. For example, $\mathbf{A}[1,2]=b$ in the above two examples. The rows and columns of matrices are normally numbered from $1$ instead of from 0; thus, $1\leq i \leq M$ and $1\leq j \leq N$. When $N=M$, the matrix is said to be square.

The transpose of a real matrix $\mathbf{A}\in{\bf R}^{M\times N}$ is denoted by $\mathbf{A}^{\!\hbox{\tiny T}}$ and is defined by

$$\mathbf{A}^{\!\hbox{\tiny T}}[i,j] \isdef \mathbf{A}[j,i].$$

While $\mathbf{A}$ is $M\times N$, its transpose is $N\times M$. We may say that the ``rows and columns are interchanged'' by the transpose operation, and transposition can be visualized as ``flipping'' the matrix about its main diagonal. For example,

$$\left[\begin{array}{cc} a & b \\ c & d \\ e & f \end{array}\right... ...\tiny T}} =\left[\begin{array}{ccc} a & c & e \\ b & d & f \end{array}\right].$$

A complex matrix $\mathbf{A}\in{\bf C}^{M\times N}$, is simply a matrix containing complex numbers. The transpose of a complex matrix is normally defined to include conjugation. The conjugating transpose operation is called the Hermitian transpose. To avoid confusion, in this tutorial, $\mathbf{A}^{\!\hbox{\tiny T}}$ and the word ``transpose'' will always denote transposition without conjugation, while conjugating transposition will be denoted by $A^{\ast }$ and be called the ``Hermitian transpose'' or the ``conjugate transpose.'' Thus,

$$A^{\ast }[i,j] \isdef \overline{\mathbf{A}[j,i]}.$$