(Matrix)
With m , n ∈ R m,n \in \mathbb{R} m , n ∈ R ( m , n ) (m,n) ( m , n ) matrix A \mathbf{A} A m . n m.n m . n a i j , i = 1 , … , m , j = 1 … , n a_{ij}, i=1,\dots,m,j=1\dots,n a ij , i = 1 , … , m , j = 1 … , n m m m n n n
A = [ a 11 a 12 … a 1 n a 21 a 22 … a 2 n ⋮ ⋮ ⋮ a m 1 a m 2 … a m n ] , a i j ∈ R . \begin{align}
\mathbf{A} = \begin{bmatrix}
a_{11}&a_{12}&\dots &a_{1n}\\
a_{21}&a_{22}&\dots &a_{2n}\\
\vdots&\vdots&&\vdots \\
a_{m1}&a_{m2}&\dots&a_{mn}
\end{bmatrix}, a_{ij} \in \mathbb{R}. \tag{2.11}
\end{align} A = a 11 a 21 ⋮ a m 1 a 12 a 22 ⋮ a m 2 … … … a 1 n a 2 n ⋮ a mn , a ij ∈ R . ( 2.11 ) (1, n n n rows and (m m m columns . These are also called row/column vectors .
(Matrix Operations)
Addition
The sum of two matrices A ∈ R m × n , B ∈ R m × n \mathbf{A} \in \mathbb{R}^{m\times n}, \mathbf{B} \in \mathbb{R}^{m\times n} A ∈ R m × n , B ∈ R m × n
A + B ≔ [ a 11 + b 11 … a 1 n + b 1 n ⋮ ⋮ ⋮ a m 1 + b m 1 … a m n + b m n ] ∈ R m × n . \begin{align}
\mathbf{A}+\mathbf{B} \coloneqq
\begin{bmatrix}
a_{11}+b_{11} &\dots& a_{1n}+b_{1n} \\
\vdots&\vdots&\vdots \\
a_{m1}+b_{m1}&\dots&a_{mn}+b_{mn}
\end{bmatrix} \in \mathbb{R}^{m\times n}. \tag{2.12}
\end{align} A + B : = a 11 + b 11 ⋮ a m 1 + b m 1 … ⋮ … a 1 n + b 1 n ⋮ a mn + b mn ∈ R m × n . ( 2.12 ) Multiplication
For matrices A ∈ R m × n , B ∈ R n × k \mathbf{A} \in \mathbb{R}^{m\times n}, \mathbf{B} \in \mathbb{R}^{n\times k} A ∈ R m × n , B ∈ R n × k c i j c_{ij} c ij C = A B ∈ R m × n \mathbf{C} = \mathbf{A}\mathbf{B} \in \mathbb{R}^{m \times n} C = AB ∈ R m × n
c i j = ∑ l = 1 n a i l b l j , i = 1 , … , m , j = 1 , … , k . \begin{align}
c_{ij} = \sum_{l=1}^n a_{il}b_{lj}, \;\;i=1,\dots,m, \;j=1,\dots, k. \tag{2.13}
\end{align} c ij = l = 1 ∑ n a i l b l j , i = 1 , … , m , j = 1 , … , k . ( 2.13 ) (Identity Matrix)
In R n × n \mathbb{R}^{n\times n} R n × n identity matrix as a square matrix which
I n ≔ [ 1 0 … 0 … 0 0 1 … 0 … 0 ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ 0 0 … 1 … 0 ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ 0 0 … 0 … 1 ] ∈ R n × n \begin{align}
\mathbf{I}_{n} \coloneqq \begin{bmatrix}
1&0&\dots&0&\dots&0 \\
0&1&\dots&0&\dots&0 \\
\vdots&\vdots&\ddots&\vdots&\ddots&\vdots \\
0&0&\dots&1&\dots&0 \\
\vdots&\vdots&\ddots&\vdots&\ddots&\vdots \\
0&0&\dots&0&\dots&1
\end{bmatrix} \in \mathbf{R}^{n\times n} \tag{2.17}
\end{align} I n : = 1 0 ⋮ 0 ⋮ 0 0 1 ⋮ 0 ⋮ 0 … … ⋱ … ⋱ … 0 0 ⋮ 1 ⋮ 0 … … ⋱ … ⋱ … 0 0 ⋮ 0 ⋮ 1 ∈ R n × n ( 2.17 ) contains 1 1 1 0 0 0
Properties
∀ A ∈ R m × n , B ∈ R n × p , C ∈ R p × q : ( A B ) C = A ( B C ) \begin{align}
\forall\mathbf{A}\in\mathbb{R}^{m \times n},\mathbf{B}\in\mathbb{R}^{n \times p}, \mathbf{C}\in\mathbb{R}^{p \times q} : (\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C}) \tag{2.18}
\end{align} ∀ A ∈ R m × n , B ∈ R n × p , C ∈ R p × q : ( AB ) C = A ( BC ) ( 2.18 ) ∀ A , B ∈ R m × n , C , D ∈ R n × p : ( A + B ) C = A C + B C A ( C + D ) = A C + A D \forall \mathbf{A}, \mathbf{B}\in\mathbb{R}^{m \times n}, \mathbf{C},\mathbf{D}\in\mathbb{R}^{n \times p}: \begin{align*}
(\mathbf{A}+\mathbf{B})\mathbf{C} = \mathbf{A}\mathbf{C}+\mathbf{B}\mathbf{C} \\
\mathbf{A}(\mathbf{C}+\mathbf{D}) = \mathbf{A}\mathbf{C}+\mathbf{A}\mathbf{D}
\end{align*} ∀ A , B ∈ R m × n , C , D ∈ R n × p : ( A + B ) C = AC + BC A ( C + D ) = AC + AD
Multiplication with Identity :
∀ A ∈ R m × n : I m A = A I n = A \begin{align}
\forall \mathbf{A} \in \mathbb{R}^{m\times n}: \mathbf{I}_{m}\mathbf{A} = \mathbf{A}\mathbf{I}_{n} = \mathbf{A} \tag{2.20}
\end{align} ∀ A ∈ R m × n : I m A = A I n = A ( 2.20 ) Connections