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Section 3.1 First definitions

We start our study of matrix theory with some important definitions:

Definition 3.1.1. Matrix.

A matrix (the plural is matrices) is a rectangular array of numbers.

Here is a matrix:

\begin{equation*} \begin{bmatrix} 1\amp2\amp-1\amp4\\ 2\amp1\amp3\amp0\\ 4\amp4\amp3\amp1 \end{bmatrix} \end{equation*}

The rows are horizontal parts of the array and the columns are the vertical ones. We often denote the rows by \(R_1, R_2,\dots\) and the columns by \(C_1, C_2,\dots\text{.}\)

In the previous example we have three rows and four columns:

Rows: \(R_1=\begin{bmatrix}1\amp2\amp-1\amp4\end{bmatrix}\text{,}\) \(R_2=\begin{bmatrix}2\amp1\amp3\amp0\end{bmatrix}\) and \(R_3=\begin{bmatrix}4\amp4\amp3\amp1\end{bmatrix}\)

Columns: \(C_1=\begin{bmatrix}1\amp2\amp4\end{bmatrix}\text{,}\) \(C_2=\begin{bmatrix}2\amp1\amp4\end{bmatrix}\text{,}\) \(C_3=\begin{bmatrix}-1\amp3\amp3\end{bmatrix}\) and \(C_4=\begin{bmatrix}4\amp0\amp1\end{bmatrix}\text{.}\)

Sometimes the columns are written vertically:

\begin{equation*} C_1=\begin{bmatrix}1\\2\\4\end{bmatrix}, C_2=\begin{bmatrix}2\\1\\4\end{bmatrix}, C_3=\begin{bmatrix}-1\\3\\3\end{bmatrix} \textrm{and } C_4=\begin{bmatrix}4\\0\\1\end{bmatrix}. \end{equation*}
Definition 3.1.2. Size of a matrix.

If a matrix had \(m\) rows and \(n\) columns, then we say that the matrix has size \(m\times n\text{.}\) We call this an \(m\) by \(n\) matrix. When it is desirable to emphasize the size of a matrix \(A\text{,}\) the following notation is used:

\begin{equation*} A= \begin{bmatrix} a_{1,1} \amp \cdots \amp a_{1,n}\\ \amp\vdots\amp\\ a_{m,1} \amp \cdots \amp a_{m,n} \end{bmatrix}_{m,n} \end{equation*}

Notice the special notation used: \(a_{i,j}\) is the one and only entry in the matrix that is both in the \(i\)-th row and in the \(j\)-th column.

Sometimes \(m\times n\) is called the shape of the matrix.

Definition 3.1.3. Square matrix.

An \(m\times n\) matrix is called square if \(m=n.\) To emphasize the size of a square matrix, a single subscript is may be used:

\begin{equation*} A= \begin{bmatrix} a_{1,1} \amp \cdots \amp a_{1,n}\\ \amp\vdots\amp\\ a_{n,1} \amp \cdots \amp a_{n,n} \end{bmatrix}_n \end{equation*}
Definition 3.1.4. Equality of matrices.

Two matrices \(A=[a_{i,j}]\) and \(B = [b_{i,j}]\) are equal if they are the same size, say \(m\) by \(n,\) and

\begin{equation*} a_{i,j}=b_{i,j}, \text{ for } i=1,2,\dots,m \text{ and } j=1,2,\dots,n. \end{equation*}

In other words, they matrices are equal entry-wise.

Example 3.1.5. Equal matrices.

  1. As a first example, consider

    \begin{equation*} \begin{bmatrix} 1 \amp 2 \amp 1\\ 1 \amp -1 \amp 0 \end{bmatrix} = \begin{bmatrix} 5-4 \amp 12/6 \amp \cos(0)\\ 10^0 \amp (-1)^5 \amp \sin(4\pi) \end{bmatrix} \end{equation*}

    There are six entries in these matrices. We need to check all six to establish equality; they are

    \begin{align*} 1\amp=5-4\\ 2\amp=\frac{12}6\\ 1\amp=\cos(0)\\ 1\amp=10^0\\ -1\amp=(-1)^5\\ 0\amp=\sin(4\pi) \end{align*}

    Since all of the individual equations are valid, the two matrices are equal.

  2. Matrix equality can encode interesting information.

    \begin{equation*} \begin{bmatrix} x+y \amp x+2y\\ 2x+y \amp 2x+2y \end{bmatrix} = \begin{bmatrix} 2 \amp 3 \\ 3 \amp 4 \end{bmatrix} \end{equation*}

    (note that this is the same as four equations in two unknowns; they imply \(x=y=1\))

On the face of it, in order to show that two \(m\times n\) matrices are equal, we must check the equality of all \(mn\) entries. One reason we study mathematics is to make calculations easier. Our mathematical development will, in fact, do just that.

There are two matrices that appear so often that they have special names:

Definition 3.1.6. The zero matrix.

The zero matrix, written as \(\vec0\text{,}\) has every entry equal to \(0\text{.}\)

\begin{equation*} \vec0= \begin{bmatrix} 0 \amp 0\amp \cdots \amp 0\\ 0 \amp 0\amp \cdots \amp 0\\ \amp \amp \vdots \amp \\ 0 \amp 0\amp \cdots \amp 0 \end{bmatrix} \end{equation*}
Definition 3.1.7. The identity matrix.

The identity matrix, written as \(I\text{,}\) is a square matrix in which every entry equal to \(0\) or \(1\text{.}\)

\begin{equation*} I= \begin{bmatrix} 1 \amp 0\amp \cdots \amp 0 \amp 0\\ 0 \amp 1\amp \cdots \amp 0 \amp 0\\ \amp \amp \vdots \amp \\ 0 \amp 0\amp \cdots \amp 1 \amp 0\\ 0 \amp 0\amp \cdots \amp 0 \amp 1 \end{bmatrix} \end{equation*}

When we want to emphasize the size of the matrix is \(n\times n\text{,}\) we write it as \(I_n\text{.}\) Hence, if \(I_n=[a_{i,j}]\text{,}\) we have

\begin{equation*} a_{i,j}= \begin{cases} 1 \amp \textrm{if } i=j\\ 0 \amp \textrm{if } i\not=j \end{cases} \end{equation*}