A zero matrix has every entry equal to zero: \(\textbf{0} = [0]\) or \(0 = [0]\) , or explicitly

\[\begin{align} \textbf{0} = \begin{bmatrix} 0 & 0 &\cdots & 0 \\ 0 & 0 &\cdots & 0 \\ \cdots & \cdots &\cdots & \cdots \\ & & & \\ \cdots & \cdots &\cdots & \cdots \\ 0 & 0 &\cdots & 0 \\ \end{bmatrix} \end{align}\]

An identity matrix is a square matrix whose diagonal entries are all equal to one and whose off-diagonal entries are all equal to zero, and is denoted by \(I\) or \(\textbf{I}\).

\[\begin{align} \textbf{0} = \begin{bmatrix} 1 & 0 &\cdots & 0 \\ 0 & 1 &\cdots & 0 \\ \cdots & \cdots &\cdots & \cdots \\ & & & \\ \cdots & \cdots &\cdots & \cdots \\ 0 & 0 &\cdots & 1 \\ \end{bmatrix} \end{align}\]

People also use the following natation

• \[\begin{align} \textbf{I} = \begin{bmatrix} \delta_{ij} \\ \end{bmatrix} \end{align}\]

where \(\begin{cases} {1 \quad \text{when} \quad i = j \\ 0 \quad \text{ otherwise }}\end{cases}\)

is called the Kronecker delta.

A square matrix in which all the entries off the main diagonal are zero. Here are some examples:

• \[\begin{align} & \begin{bmatrix} 1 & 0 \\ 0 & 2 \\ \end{bmatrix} & & \begin{bmatrix} e & 0 & 0 \\ 0 & \frac{1}{2} & 0\\ 0 & 0 & 4 \end{bmatrix} \end{align}\]

Row vector: A row vector is of dimension \(p\) is a \(1 \times p\) matrix: \[\begin{align} u &= \begin{bmatrix} u_{1} & u_{2} & \cdots & u_{p} \\ \end{bmatrix} \end{align}\]

Column vector: A column vector of dimension n is an n × 1 matrix:

\[\begin{align} v &= \begin{bmatrix} v_{1} \\ v_{2} \\ \vdots \\ v_{p} \\ \end{bmatrix} \end{align}\]

The transpose of a column vector becomes a row vector, and vice versa.

If \(\textbf{A}^{t} = \textbf{A}\) or \(a_{ij} = a_{ji}\), then the matrix \(\textbf{A}\) is said to be symmetric.

Of course, a symmetric matrix must be a square matrix.

The sum/difference of two matrices \(\textbf{A}\) and \(\textbf{B}\) is defined by the sum/difference of corresponding entries, i.e.,\(\textbf{A} + \textbf{B} = [a_{ij} + b_{ij}]\).

The equation of \(\textbf{A} = \textbf{B}\) is equivalent to \(\textbf{A} - \textbf{B} = 0\).

If \(A\) is any matrix and \(c\) is any scalar, then the product \(cA\) is the matrix obtained by multiplying each entry of the matrix \(A\) by \(c\). The matrix \(cA\) is said to be a scalar multiple of \(A\).

In matrix notation, if \(A = [a_{ij}]\), then \[(cA)_{ij} = c(A)_{ij} = ca_{ij}\]

If \(A\) is a square matrix, then the trace of A, denoted by \(tr(A)\), is defined to be the sum of the entries on the main diagonal of \(A\). The trace of \(A\) is undefined if \(A\) is not a square matrix.

The amplitude of vector u of dimension n is defined as \[|\textbf{u}| = \sqrt{{u_{1}}^{2} + {u_{2}}^{2} + \cdots {u_{n}}^{2}}\].

Sometimes, the amplitude is also called length, or Euclidean length, or magnitude.

If the Euclidean length of \(\textbf{u}\) is equal to one, we say that the \(\textbf{u}\) is a unit vector. If every element of \(\textbf{u}\) is zero, then we say that \(\textbf{u}\) is a zero vector.

By the definition of dot product, we have \[|\textbf{u}|^{2} = \textbf{u} \cdot \textbf{u}\].

If \(\textbf{u}\cdot \textbf{v} = 0\), we say that u and v are orthogonal. Further, if \(\textbf{u}\cdot \textbf{v} = 0\) and \(\textbf{u}=\textbf{v} = 1\), then we say that \(\textbf{u}\) and \(\textbf{v}\) are orthonormal.

Let \(A\) be an \(m \times n\) matrix and \(B\) an \(n \times p\) matrix: then the product \(AB\) is an \(m \times p\). The \(ij\) term of \(AB\) is the dot product of the ith row vector of \(A\) with the jth column vector of \(B\), so that

\[\begin{align} (AB)_{ij} &= a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj}\\ &= \sum_{i=1}^{n} a_{ik}b_{kj} \end{align}\]

If \(A_{1}, A_{2}, \cdots A_{r}\) are matrices of the same size, and if \(c_{1}, c_{2}, \cdots c_{r}\) are scalars, then an expression of the form

\[c_{1}A_{1} + c_{2}A_{2} + \cdots + c_{r}A_{r}\] with coefficients \(c_{1}, c_{2}, \cdots c_{r}\)

is called a linear combination of \(A_{1}, A_{2}, \cdots A_{r}\)

A linear equation in the variables \(x_{1}, \cdots, x_{n}\) is an equation that can be written in the form \[a_{1}x_{1} + a_{2}x_{2} + \cdots + a_{n}x_{n} = b\] where \(b\) and the coefficients \(a_{1}, \cdots, a_{n}\) are real or complex numbers, usually known in advance.

The subscript \(n\) may be any positive integer.

A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables–say, \(x_{1}, \cdots, x_{n}\). An example is:

\[\begin{align} 2x - y + 1.5z &= \phantom{-}8\\ x \quad \quad \, -4z &= -7 \end{align}\]

A solution of the system is a list \((s_{1}, \cdots, s_{n})\) of numbers that makes each equation a true statement when the values \(s_{1}, \cdots, s_{n}\) are substituted for \(x_{1}, \cdots, x_{n}\), respectively.

The set of all possible solutions is called the solution set of the linear system.

Two linear systems are called equivalent if they have the same solution set.

A system of linear equations has

• no solution, or
• exactly one solution, or
• infinitely many solutions.

A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.

The essential information of a linear system can be recorded compactly in a rectangular array called a matrix. Given the system

\[\begin{align} x - 2y \,\, + z &= \, \, 0\\ \quad \, 2y - 8z &= \, \, 8\\ 5x \quad \quad -5z &= 10\\ \end{align}\]

with the coefficients of each variable aligned in columns, the matrix

\[\begin{align} \begin{bmatrix} 1 & -2 & \phantom{-}1\\ 0 & \phantom{-}2 & -8\\ 5 & \phantom{-}0 & -5 \end{bmatrix} \end{align}\]

is called the coefficient matrix (or matrix of coefficients) of the system and the matrix

\[\begin{align} \begin{bmatrix} 1 & -2 & \phantom{-}1 & 0\\ 0 & \phantom{-}2 & -8 & 8\\ 5 & \phantom{-}0 & -5 & 10 \end{bmatrix} \end{align}\] is called the augmented matrix of the system.

Solve the above system.

• (Replacement) Replace one row by the sum of itself and a multiple of another row.

• (Interchange) Interchange two rows.

• (Scaling) Multiply all entries in a row by a nonzero constant.

\(\color{forestgreen}{\textbf{Solution | }}\) The results agree with the right side of the original system, so \((1, 0, -1)\) is a solution of the system.