Introduction to Matrices
The matrix of type \(m \times n\) is a scheme of m times n elements, for example real or complex numbers, functions, differential quotients, vectors etc., that are arranged in \(m\) rows and \(n\) columns.
\begin{eqnarray}
\mathbf{A} &=& (a_{ij}) =
\left(
\begin{matrix}
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{matrix}
\right)
\end{eqnarray}
The number of rows \(m\) and columns \(n\) of that matrix are completely determined by the type of matrix. If \(m = n\), we are talking about a square matrix, and otherwise about a rectangular matrix.
Real and Complex matrix
The real matrix have elements that are real numbers, and complex matrix is type of matrix that consist of complex numbers which can be written in general form as:
\begin{eqnarray}
a_{ij} + ib_{ij}.
\end{eqnarray}
The complex matrix can be written as two real matrix \(\mathbf{A}\) and \(\mathbf{B}\) in a following form:
\begin{eqnarray}
\mathbf{A} + i\mathbf{B}.
\end{eqnarray}
If \(\mathbf{A}\) is complex matrix, than the complex conjugate matrix \(\overline{\mathbf{A}}\) with elements:
\begin{eqnarray}
\overline{a}_{ij} &=& Re(a_{ij}) - i Im(a_{ij}).
\end{eqnarray}
Adjoint Matrix
For complex matrix \(\mathbf{A}\), their adjoint matrix \(\mathbf{A}^H\) is obtained by transposing the conjugated matrix \(\overline{\mathbf{A}}\). The most common symbol for adjoint matrix is \(\mathbf{A}*\).
\begin{eqnarray}
\mathbf{A}* &=& \mathbf{A}^H = (\overline{A})^T.
\end{eqnarray}
Null matrix
The null matrix is type of matrix with all its elements equal to 0.
\begin{eqnarray}
\mathbf{A_0} &=& \left(\begin{matrix}0&0&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0\\\end{matrix}\right)
\end{eqnarray}
Square matrices
The square matrices are type of matrices in which number of rows is equal to the number of columns, i.e. \(m=n\):
\begin{eqnarray}
\mathbf{A} &=& \mathbf{A}_{nn} = \left(\begin{matrix}a_{11}&\cdots&a_{1n}\\ \vdots& \ddots & \vdots\\ a_{n1} & \cdots & a_{nn}\end{matrix}\right)
\end{eqnarray}
The main diagonal of matrix \(\mathbf{A}\) consist of members \(a_{ij}\) for which \(i=j\) is valid i.e. these are elements \(a_{11}, a_{22},...,a_{nn}\). This diagonal goes from the upper left to the right lower member of the matrix.
Diagonal matrix
Diagonal matrix \(\mathbf{D}\) is square matrix in which all elements outside of main diagonal are equal to zero. The general form of diagonal matrix can be written as:
\begin{eqnarray}
\mathbf{D} &=& \left(\begin{matrix}a_{11}&0&\cdots&0\\ 0&a_{22}&\cdots& 0 \\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&a_{nn}\end{matrix}\right).
\end{eqnarray}
Scalar matrix S
The scalar matrix S is type of diagonal matrix in which all members on main diagonal are equal for example to some real or complex number:
\begin{cases}
a_{ij} = 0 & \mbox{if } i\neq j,\\
a_{ii} = c & \mbox{for every } i.
\end{cases}
Trace Matrix
The trace of matrix is the sum of all diagonal matrix members and can be written as:
\begin{eqnarray}
\mathrm{Tr}(\mathbf{A}) &=& a_{11} + a_{22} + \cdots + a_{nn} = \sum_{i=1}^{n} a_{ii}.
\end{eqnarray}
Symmetric Matrices
The square matrix \(\mathbf{A}\) is symmetric matrix if it is equal to its transpose matrix. The relation in mathematical form can be written as:
\begin{eqnarray}
\mathbf{A} &=& \mathbf{A}^T.
\end{eqnarray}
The members if diagonal matrix are equal with respect to mirroring over the main diagonal, i.e.:
\begin{eqnarray}
a_{ij} &=& a_{ji}.
\end{eqnarray}
Normal matrix
A normal matrix is type of matrix for which the following relation is valid:
\begin{eqnarray}
\mathbf{A}^T \mathbf{A} &=& \mathbf{A}\mathbf{A}^T.\\
\end{eqnarray}
So the normal matrix is type of matrix for which the product value of transpose and original matrix is equal regardless of the order of multiplication.
Antisymmetric matrix
The antisymmetric matrix is square matrix \(\mathbf{A}\) for which the following relation is valid:
\begin{eqnarray}
\mathbf{A}&=& -\mathbf{A}^T.\\
\end{eqnarray}
For members \(a_{ij}\) of antisymmetric matrix the follwing relation is valid:
\begin{eqnarray}
a_{ij} &=& a_{ji}, \quad a_{ii} = 0.
\end{eqnarray}
The trace of antisymmetric matrix vanishes:
\begin{eqnarray}
\mathrm{Tr}(\mathbf{A}) &=& 0.
\end{eqnarray}
The mirror terms with respect to the main diagonal of such a matrix differ only in sign. Each square matrix \(\mathbf{A}\) can be decomposed into the sum of one symmetric matrix \(\mathbf{A_s}\) and one antisymmetric matrix \(\mathbf{A_{as}}\):
\begin{eqnarray}
\mathbf{A} &=& \mathbf{A_s} + \mathbf{A_{as}}\\
\mathbf{A_s} &=& \frac{1}{2}(\mathbf{A}+\mathbf{A}^T),\\
\mathbf{A_{as}} &=& \frac{1}{2}(\mathbf{A}-\mathbf{A}^T).
\end{eqnarray}
Hermitian or slef-adjoint matrices
The square matrix \(\mathbf{A}\) is hermitian or slef-adjoint matrix, if it is equal to its own adjoint matrix:
\begin{eqnarray}
\mathbf{A} &=& (\mathbf{A})^T = \mathbf{A}^* = \mathbf{A}^H.
\end{eqnarray}
Real matrix is hermitian if and only if it is symmetric. Determinant of hermitian matrix is real number.
Anti-hermitian matrix
The square matrix is anti-hermitian matrix, if it is equal to its own negative adjoint matrix:
\begin{eqnarray}
\mathbf{A} &=& (\overline{\mathbf{A}})^T = -\mathbf{A}^* = -\mathbf{A}^H.
\end{eqnarray}
For members \(a_{ij}\) and for the trace of antihermitian matrix the following relations are valid:
\begin{eqnarray}
a_{ij} &=& -\overline{a_{ji}}^*, \quad a_{ii} = 0, \quad \mathrm{Tr}(\mathbf{A}) = 0.
\end{eqnarray}
Every square matrix \(\mathbf{A}\) can be decomposed into the sum of one Hermitian matrix \(\mathbf{A_{h}}\) and one antihermitian matrix \(\mathbf{A_{ah}}\):
\begin{eqnarray}
\mathbf{A} &=& \mathbf{A_h} + \mathbf{A_{ah}},\\
\mathbf{A_h} &=& \frac{1}{2}(\mathbf{A}+ \mathbf{A}^*),\\
\mathbf{A_{ah}} &=& \frac{1}{2}(\mathbf{A} - \mathbf{A}^*).
\end{eqnarray}
Unit matrices I
The unit matrix is every matrix in which all members on the diagonal are equal to 1, and every other member is equal to 0. The unit matix can be written as:
\begin{eqnarray}
I &=& \left(\begin{matrix}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots& \ddots& \vdots\\ 0&0&\cdots&1\end{matrix}\right) = \delta_{ij}\quad \mbox{where } \delta_{ij} = \begin{cases}0& \mbox{for } i\neq j,\\ 1 &\mbox{for } i = j. \end{cases}
\end{eqnarray}
The symbol introduced in previous equation \(\delta_{ij}\) is Kroneckerov delta symbol.
Traingular matrix
There are two types of trinagular matrices and these are upper and lower traingular matrices. The upper or right traingular matrix \(\mathbf{U}\) is square matrix where all members below the main diagonal are equal to zero.
\begin{eqnarray}
\mathbf{U} &=& (a_{ij}) \quad \mathrm{where } a_{ij} = 0 \quad \mathrm{for } j > i.
\end{eqnarray}
The lower or left traingular matrix \(\mathbf{L}\) is square matrix where all members below main diagonal are equal to zero:
\begin{eqnarray}
\mathbf{L} &=& (b_{ij}) \quad \mathrm{where } b_{ij} = 0 \quad \mathrm{for } j < i.
\end{eqnarray}
Vectors
The matrix of type \(n\times 1\) is called single column matrix or column vector with dimension \(n\), and matrix of type \(1\times n\) is called single row matrix or row vector.
\begin{eqnarray}
\mathrm{vector-column: }\mathbf{a} &=& \left(\begin{matrix}a_1\\ a_2\\ \cdots \\ a_n\end{matrix}\right),\\
\mathrm{vector-row: }\mathrm{a}^T = (a_1, a_2,...,a_n).
\end{eqnarray}
Transposing column vector we obtain row vector and vice versa.Every point of Euclidian n-dimensional space can be described with vector-column or row-vector. The null vector is denoted with \(\mathbf{o}\) or \(\mathbf{o}^T\).