Matrices

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

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

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^D.1 of a matrix $A$ may be denoted by $A[i,j]$ or $A(i,j)$. 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 $A\in{\bf R}^{M\times N}$ is denoted by $A^{\hbox{\tiny T}}$ and is defined by

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

Note that while $A$ is $M\times N$, its transpose is $N\times M$.

A complex matrix $A\in{\bf C}^{M\times N}$, 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, $A^{\hbox{\tiny T}}$ 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,

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

Example: The transpose of the general $3\times 2$ matrix is

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

while the conjugate transpose of the general $3\times 2$ matrix is

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

A column vector, e.g.,

$$\underline{x}= \left[\begin{array}{c} x_0 \\ [2pt] x_1 \end{array}\right]$$

is the special case of an $M\times 1$ matrix, and a row vector, e.g.,

$$\underline{x}^{\hbox{\tiny T}} = [x_0\; x_1]$$

(as we have been using) is a $1\times N$ matrix. In contexts where matrices are being used (only this section for this book), it is best to define all vectors as column vectors and to indicate row vectors using the transpose notation, as was done in the equation above.