Objectives
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Definitions of matrices and vectors.
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Learning some notation around matrices and vectors.
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Definitions of upper and lower triangular matrices.
Section 2.1 Matrix and Vector Definition
Subsection 2.1.1 Matrices
Definition 2.1.1.
A matrix is a rectangular grid of numbers. An \(m\) by \(n\) matrix has \(m\) rows and \(n\) columns.
For example,
\begin{equation}
\begin{bmatrix} 1 \amp 2 \amp 11 \amp -1 \\ 3 \amp -2 \amp 3 \amp 4 \end{bmatrix}\tag{2.1.1}
\end{equation}
The size of a matrix is the number of rows and columns in the matrix. The number of rows is listed first. The size of the example above is 2 by 4.
The numbers in a matrix are called the entries or elements of the matrix. For example, for the matrix in (2.1.1), the entry on the first row and third column is 11. Often we will use the notation \(a_{1,3} = 11\text{,}\) where the subscript 1 is the row number and \(3\) is the column number.
Subsection 2.1.2 Vectors
Definition 2.1.2.
A vector is a matrix with one of its dimensions is 1. If a matrix only has one column it is called a column vector. If the matrix has only one row it is called a a row vector. The number of elements (numbers) in the vector is called the length.
The following are examples of vectors.
Example 2.1.3.
The first two are column vectors. The third is a row vectors.
\begin{equation*}
\begin{aligned} \boldsymbol{u} \amp = \begin{bmatrix} 3 \\ -1 \\ 0 \\ 2 \end{bmatrix} \amp \boldsymbol{v} \amp = \begin{bmatrix} 110 \\ 25 \\ 150 \end{bmatrix} \amp \boldsymbol{w} \amp = \begin{bmatrix} 3 \amp 4 \amp -2 \end{bmatrix} \end{aligned}
\end{equation*}
Since a vector has a single dimension, the individual elements can be identified with a single index. For example, \(v_2 = -1 \) and \(w_3 = -2\text{.}\) The lengths of \(\boldsymbol{u}, \boldsymbol{v}\) and \(\boldsymbol{w}\) are 4,3 and 3 respectively.
Note 2.1.4.
Vectors can be thought of as individual columns or rows of a matrix. Also, a column vector can be thought of as an \(m \times 1\) matrix and a row vector as a \(1 \times n\) matrix and properties of matrices will pertain to vectors as well.
Subsection 2.1.3 Matrix, Vector and Scalar Notation
A scalar is a fancy term for a number. Mathematicians use this term to distinguish them from matrices and vectors, which are not scalars. Whenever variables are used for scalars, then lower case letters will be used. For example, \(a, n\) and \(x\) are scalar variables.
When we use variable names for matrices, we will use capital letters. For example, \(A, B\) and \(X\) are matrix (or vector) variables.
Generally, vectors will be lower case and bold. For example, here are vectors: \(\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{x}\text{.}\) Note: sometimes vectors are indicated with a arrow above the variable name, like \(\vec{v}\text{.}\)
Remark 2.1.5.
In general, a matrix \(A\) can be written \([a_{ij}]\text{.}\) These will be used in proofs later on in this chapter and for defintions of operations. A vector \(\boldsymbol{v}\) can be written \([v_i]\) and although this notation can be used for both column and row vectors, unless denoted as such vectors will be consider as column vectors. We will see later in this chapter that we can simply write a row vector as a transformation of a column vector.
Subsection 2.1.4 Diagonal and Triangular Matrices
As we will see throughout this chapter, there are many important forms of matrices and in this section we see diagonal and triangular matrices.
Definition 2.1.6.
A matrix with only non-zero entries are on the main diagonal is called a diagonal matrix. Using the subscript notation, \(A\) is diagonal if \(a_{ij} =0\) for all \(i,j\) where \(i \neq j\text{.}\)
1
Recall that the main diagonal is the one where the row number equals the column number, or more informally the diagonal from upper left to lower right.
The following show examples of diagonal matrices.
Example 2.1.7.
These matrices are diagonal matrices.
\begin{equation*}
\begin{aligned}
\amp\begin{bmatrix}
2 \amp 0 \amp 0 \\
0 \amp -1 \amp 0 \\
0 \amp 0 \amp 5
\end{bmatrix} \amp\amp \begin{bmatrix}
5 \amp 0 \\
0 \amp 2/3 \\
0 \amp 0 \\
0 \amp 0
\end{bmatrix} \amp\amp \begin{bmatrix}
-9 \amp 0 \amp 0 \\
0 \amp 4 \amp 0
\end{bmatrix}
\end{aligned}
\end{equation*}
Another class of matrices are triangular. These are matrices with all zeros above or below the diagonal.
Definition 2.1.8.
An upper-triangular matrix is a matrix where all entries below the main diagonal is 0 or \(A\) is upper-triangular if \(a_{ij}=0\) if \(i\gt j\text{.}\)
A lower-triangular matrix is a matrix where all entries above the main diagonal is 0 or \(A\) is lower-triangular if \(a_{ij}=0\) if \(i\lt j\text{.}\)
And here are some examples.
Example 2.1.9.
The following are upper-triangular matrices.
\begin{equation*}
\begin{aligned}
\amp \begin{bmatrix}
-1 \amp 2 \amp 0 \amp 4 \\
0 \amp 4 \amp -5 \amp 3 \\
0 \amp 0 \amp 1 \amp -2
\end{bmatrix}
\amp \amp
\begin{bmatrix}
5 \amp 3 \amp 4 \\
0 \amp -2 \amp 11 \\
0 \amp 0 \amp 3 \\
0 \amp 0 \amp 0
\end{bmatrix}
\amp \amp
\begin{bmatrix}
6 \amp 4 \amp 0 \amp 3 \\
0 \amp -2 \amp 1 \amp 5 \\
0 \amp 0 \amp 4 \amp 3 \\
0 \amp 0 \amp 0 \amp -4
\end{bmatrix}
\end{aligned}
\end{equation*}
The following are lower-triangular matrices.
\begin{equation*}
\begin{aligned}
\amp \begin{bmatrix}
3 \amp 0 \amp 0 \\
-1 \amp 2 \amp 0 \\
4 \amp 5 \amp -2
\end{bmatrix}
\amp \amp
\begin{bmatrix}
7 \amp 0 \amp 0 \amp 0 \\
2 \amp -3 \amp 0 \amp 0 \\
1 \amp 4 \amp 5 \amp 0
\end{bmatrix}
\amp \amp
\begin{bmatrix}
1 \amp 0 \amp 0 \amp 0 \\
-2 \amp 3 \amp 0 \amp 0 \\
4 \amp -1 \amp 2 \amp 0 \\
5 \amp 3 \amp -2 \amp 6
\end{bmatrix}
\end{aligned}
\end{equation*}
