Definition

$A_{m \times n} = a_{11} ⋮ a_{m 1} \dots ⋱ \dots a_{1 n} ⋮ a_{mn}$

Operations

Addition, Subtraction

$A_{m \times n} \pm B_{m \times n} = a_{11} \pm b_{11} ⋮ a_{m 1} \pm b_{m 1} \dots ⋱ \dots a_{1 n} \pm b_{1 n} ⋮ a_{mn} \pm b_{mn}$

Scalar Multiplication

$k A_{m \times n} = k a_{11} ⋮ k a_{m 1} \dots ⋱ \dots k a_{1 n} ⋮ k a_{mn}$

Transpose

$A^{⊺}$ $a_{ij} \to ⊺ a_{ji}$ , $A_{m \times n} \to ⊤ A_{n \times m}$

Trace

Determinant

Matrix Multiplication

$A_{m \times p} \cdot B_{p \times n} = a_{11} ⋮ a_{m 1} \dots ⋱ \dots a_{1 p} ⋮ a_{m p} b_{11} ⋮ b_{p 1} \dots ⋱ \dots b_{1 n} ⋮ b_{p n} = a_{11} + b_{11} + \dots + a_{1 p} + b_{p 1} ⋮ a_{m 1} + b_{11} + \dots + a_{m p} + b_{p 1} \dots ⋱ \dots a_{11} + b_{1 n} + \dots + a_{1 p} + b_{p n} ⋮ a_{m 1} + b_{1 n} + \dots + a_{m p} + b_{p n}_{m \times n}$ $A_{i}^{⊺} \cdot B_{j} = \sum_{k = 1}^{p} a_{ik} \cdot b_{kj} = c_{ij}$ where $1 \leq i \leq m$ , $1 \leq j \leq n$

Vector-Matrix Multiplication

$A x = [A_{1}, A_{2}, \dots, A_{n}] [x_{1}, x_{2}, \dots, x_{n}]^{⊺} = x_{1} A_{1} + x_{2} A_{2} + \dots + x_{n} A_{n} = \sum_{k = 1}^{n} x_{k} A_{k}$

My Knowledge Base

Explorer

Matrix