Definition 1.1.1. (Zero matrix).
We denote by \(0_n\) the \(n\times n\) matrix with all its entries zero. We call \(0_n\) the zero matrix.
Section 1.1 Basic matrix operations
In this section we denote by \(\mathbb{K}\) either the set of real numbers, \(\mathbb{R}\) or the set of complex numbers, \(\mathbb{C}\text{.}\)
Definition 1.1.2. (Identity matrix).
We denote by \(I_n\) the \(n\times n\) matrix with all its diagonal entries \(1\text{,}\) and all other (non-diagonal) entries zero. We call \(I_n\) the identity matrix.
Convention 1.1.3.
The \((i,j)\)-th entry of a matrix \(A\in M_{m\times n}(\mathbb{K})\) is the entry at the intersection of \(i\)-th row and \(j\)-th column of \(A\text{.}\)
Convention 1.1.4.
We denote the set of all \(n\times n\) matrices over \(\mathbb{K}\) by \(M_n(\mathbb{K})\text{.}\) So \(M_{n\times n}(\mathbb{K})=M_n(\mathbb{K})\text{.}\) A matrix in \(M_n(\mathbb{K})\) is called a square matrix and \(n\) is called the size of a matrix.
Convention 1.1.5.
For any matrix \(A\in M_{m\times n}(\mathbb{K})\) and \(\alpha\in\mathbb{K}\) we denote by \(\alpha A\in M_{m\times n}(\mathbb{K})\) the matrix obtained by multiplying every entry of the matrix \(A\) by \(\alpha\text{.}\)
Definition 1.1.6.
A matrix in \(M_{1\times n}(\K)\) is called a column vector in \(n\)-dimensional space \(\K^n\).
A matrix in \(M_{n\times 1}(\K)\) is called a row vector in \(n\)-dimensional space \(\K^n\).
We identify \(\K\) with \(M_{1\times 1}(\K)\text{.}\)
Example 1.1.7.
\(\begin{pmatrix}1\\0\\0\\0\end{pmatrix}\) is a column vector in \(4\)-dimensional space \(\R^4\) while \(\begin{pmatrix}1\amp -i\amp 2i\end{pmatrix}\) is a row vector in \(3\)-dimensional space \(\C^3\text{.}\)
Definition 1.1.8.
Matrices \(A,B\in M_{m\times n}(\mathbb{K})\) are equal or same if every \((i,j)\)-th entry of \(A\) is equal to (same as) the \((i,j)\)-th entry of \(B\) for every \(1\leq i\leq m\) and for every \(1\leq j\leq n\text{.}\) That is, if
\begin{equation*}
A=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{m1}\amp a_{m2}\amp\cdots\amp a_{mn}\end{pmatrix}\in M_{m\times n}(\mathbb{K})\quad\text{and}\quad B=\begin{pmatrix}b_{11}\amp b_{12}\amp\cdots\amp b_{1n}\\b_{21}\amp b_{22}\amp\cdots\amp b_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\b_{m1}\amp b_{m2}\amp\cdots\amp b_{mn}\end{pmatrix}\in M_{m\times n}(\mathbb{K})
\end{equation*}
then,
\begin{equation*}
A=B\quad\text{if and only if}\quad a_{ij}=b_{ij}\,\text{for all}\,1\leq i\leq m\,\text{and}\,1\leq j\leq n.
\end{equation*}
Matrices \(A,B\in M_{m\times n}(\mathbb{K})\) are said to be not equal, and written as \(A\neq B\text{,}\) if there is at least one \(i\) and at least one \(j\) such that \((i,j)\)-th entry of \(A\text{,}\) \(a_{ij}\) and \((i,j)\)-th entry of \(B\text{,}\) \(b_{ij}\) are different, i.e., \(a_{ij}\neq b_{ij}\) (for some \(i\) and some \(j\)).
Observation 1.1.9.
Suppose that \(A\in M_{m\times n}(\mathbb{K})\) and \(B\in M_{n\times\ell}(\mathbb{K})\text{.}\) We denote the \((i,j)\)-th entry (Convention 1.1.3) \(A\) (respectively, \(B\)) by \(a_{ij}\) (respectively, \(b_{ij}\)). The \((i,j)\)-th entry of the matrix multiplication \(AB\) is given by the following formula.
\begin{equation*}
\sum_{k=1}^{n}a_{ik}b_{kj}
\end{equation*}
where \(1\leq i\leq m\) and \(1\leq j\leq\ell\text{.}\)
Definition 1.1.10. (Transpose of a matrix).
For a matrix \(A\in M_{m\times n}(\mathbb{K})\text{,}\) the transpose of \(A\text{,}\) denoted by \(A^t\text{,}\) is an \(n\times m\) matrix obtained by writing columns of \(A\) as rows of \(A^t\text{.}\) Thus if
\begin{equation*}
A=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{m1}\amp a_{m2}\amp\cdots\amp a_{mn}\end{pmatrix}\in M_{m\times n}(\mathbb{K})
\end{equation*}
then the transpose of \(A\) is
\begin{equation*}
A^t=\begin{pmatrix}a_{11}\amp a_{21}\amp\cdots\amp a_{m1}\\a_{12}\amp a_{22}\amp\cdots\amp a_{m2}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{1n}\amp a_{2n}\amp\cdots\amp a_{mn}\end{pmatrix}\in M_{n\times m}(\mathbb{K}).
\end{equation*}
Observation 1.1.11.
If \(A\in M_{m\times n}(\mathbb{K})\) then we have the following.
\begin{equation*}
\left(A^t\right)^t=A\quad\text{and}\quad\left(\alpha A\right)^t=\alpha\,A^t.
\end{equation*}
Furthermore, for \(A\in M_{\ell\times m}(\mathbb{K})\) and \(B\in M_{m\times n}(\mathbb{K})\) we have the following.
\begin{equation*}
\left(AB\right)^t=B^tA^t\in M_{n\times\ell}(\mathbb{K})
\end{equation*}
Definition 1.1.12. (Inverse of a matrix).
Let \(A\in M_n(\mathbb{K})\text{.}\) A square matrix \(B\in M_n(\mathbb{K})\) is said to be the inverse of \(A\) if
\begin{equation*}
AB=BA=I_n.
\end{equation*}
If inverse of \(A\) exists then it is denoted by \(A^{-1}\text{,}\) and \(A\) is said to be invertible.
Note 1.1.13.
Note that if \(A\in M_n(\mathbb{K})\) is invertible then the inverse of \(A^{-1}\in M_n(\mathbb{K})\) is \(A\text{,}\) i.e.,
\begin{equation*}
\left(A^{-1}\right)^{-1}=A.
\end{equation*}
Definition 1.1.14. (Coefficient matrix of a system of linear equations and homogeneous system of linear equations).
Consider a system of linear equations with \(a_{ij},b_i\in\mathbb{K}\) for every \(1\leq i\leq m\) and \(1\leq j\leq n\text{.}\)
\begin{align*}
a_{11}x_1+a_{12}x_2+ \cdots +a_{1n}x_n\amp= b_1\\
a_{21}x_1+a_{22}x_2+ \cdots +a_{2n}x_n\amp= b_2\\
\vdots\amp {}\\
a_{m1}x_1+a_{m2}x_2+ \cdots +a_{mn}x_n\amp= b_m
\end{align*}
We may write the above system of linear equations in the matrix form.
\begin{equation*}
\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{m1}\amp a_{m2}\amp\cdots\amp a_{mn}\end{pmatrix}\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}=\begin{pmatrix}b_1\\b_2\\\vdots\\b_m\end{pmatrix}
\end{equation*}
If we put
\begin{equation*}
A=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{m1}\amp a_{m2}\amp\cdots\amp a_{mn}\end{pmatrix}
\end{equation*}
and
\begin{equation*}
X=\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}\text{,}
\end{equation*}
and
\begin{equation*}
B=\begin{pmatrix}b_1\\b_2\\\vdots\\b_m\end{pmatrix}
\end{equation*}
then, the above system of linear equations can be written as
\begin{equation*}
AX=B.
\end{equation*}
The matrix \(A\) is called the coefficient matrix of the above system.
If all \(b_1=b_2=\cdots=b_m=0\) then the above system of linear equations is said to be homogeneous.
