1. What is Matrix?
a list of number of $m$ rows and $n$ column formed by $m \times n$ number of $a_{ij}$ $$ \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$ is called an $m \times n$ matrix, the numbers are called the element,$a_{ij}$ called the element located on row $i$ column $j$ if $\mathbf{A}$, $\mathbf{B}$ has same column and row, we call them the same type of matrix.2. Types of matrix
- Zero Matrix: matrix with all of its element zeroed is called a zero matrix. denote as $\mathbf{O}_{m \times n}$
- Square Matrix: when $m=n$, we call the matrix $\left.A_{n\times n}=\left(
\begin{array}
{cccc}{a_{11}} & {a_{12}} & {\cdots} & {a_{1n}} \\
{a_{21}} & {a_{22}} & {\cdots} & {a_{2n}} \\
{\vdots} & {\vdots} & & {\vdots} \\
{a_{n1}} & {a_{n2}} & {\cdots} & {a_{nn}}
\end{array}\right.\right)$ a square matrix of order $n$. Hereby, from $a_{11}$ to $a_{mn}$ is called
the main diagonal of the matrix. $a_{1n}$ to $a_{n1}$ is called the anti-diagonal of matrix.
The sum of the main diagonal of the matrix is called the trace of the matrix, denoted as $\mathrm{tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii}$ - Upper/Lower Triangular Matrix (Square Matrix): All zeros up/down the main diagonal of the matrix. $\left(\begin{array}{cccc}{a_{11}}&{0}&{\cdots}&{0}\\ {a_{21}}&{a_{22}}&{\cdots}&{0}\\ {\vdots}&{\vdots}&&{\vdots}\\ {a_{n1}}&{a_{n2}}&{\cdots}&{a_{nn}}\end{array}\right)$ is a lower triangular matrix, while $\left(\begin{array}{cccc}{a_{11}}&{a_{12}}&{\cdots}&{a_{1n}}\\ {0}&{a_{22}}&{\cdots}&{a_{2n}}\\ {\vdots}&{\vdots}&&{\vdots}\\ {0}&{0}&{\cdots}&{a_{nn}}\end{array}\right)$ is an upper triangular matrix
- Diagonal Matrix (Square Matrix): the only non-zero element in diagonal matrix is on the diagonal, this is called the diagonal matrix, denoted by $\Lambda = \left(\begin{array}{cccc}{a_{11}}&{0}&{\cdots}&{0}\\ {0}&{a_{22}}&{\cdots}&{0}\\ {\vdots}&{\vdots}&&{\vdots}\\ {0}&{0}&{\cdots}&{a_{nn}}\end{array}\right)$ or $\mathrm{diag}(a_{11},a_{22},...,a_{nn})$
- Identity Matrix (Diagonal Matrix): A diagonal matrix with all of its non-zero element being 1, denoted as $I_{n}={\left(\begin{array}{llll}{1}&{0}&{\cdots}&{0}\\ {0}&{1}&{\cdots}&{0}\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {0}&{0}&{\cdots}&{1}\end{array}\right)}$
- Scalar Matrix (Diagonal Matrix): A diagonal matrix with all of its non-zero element being $k$
- Symmetrix Matrix (Square Matrix): Matrix that satisfies $a_{ij} = a_{ji}$
- Skew-symmetric Matrix (Square Matrix): Matrix that satisfies $a_{ij} = -a_{ji}$
3. Row Echelon Form, Reduced Row Echelon Form and Normal Form
Row Echelon form are matrices that looks like $\left(\begin{array}{ccccc}{1}&{a_{0}}&{a_{1}}&{a_{2}}&{a_{3}}\\ {0}&{0}&{2}&{a_{4}}&{a_{5}}\\ {0}&{0}&{0}&{1}&{a_{6}}\\ {0}&{0}&{0}&{0}&{0}\end{array}\right)$, reduced row echelon form are matrices that looks like $\begin{array}{cccc}{\left(\begin{array}{cccc}{1}&{0}&{a_{1}}&{0}&{b_{1}}\\ {0}&{1}&{a_{2}}&{0}&{b_{2}}\\ {0}&{0}&{0}&{1}&{b_{3}}\end{array}\right)}\end{array}$Note: Column Echelon form and Reduced Column Echelon form do exist and has basically the same definition.
A matrix such that it has an identity matrix in the upleft corner and 0 for the rest is called the normal form of the matrix, looks like $\left(\begin{array}{ll}{I_{r}}&{0}\\ {0}&{0}\end{array}\right)$