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Matrix Multiplication

Matrix Multiplication
Timur2011-03-28 13:12:07

Calculator computes the product of two matrices. Some theory on the topic is placed below the calculator.

Matrix MultiplicationCreative Commons Attribution/Share-Alike License 3.0 (Unported)
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For those who forgot, The product C of two matrices A(m \times n) and B(n \times q) is defined as:
A = \begin{bmatrix} 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{bmatrix},\;\;\; B = \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1q} \\ b_{21} & b_{22} & \cdots & b_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{nq} \end{bmatrix}.

C = A \times B = \begin{bmatrix} c_{11} & c_{12} & \cdots & c_{1q} \\ c_{21} & c_{22} & \cdots & c_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ c_{m1} & c_{m2} & \cdots & c_{mq} \end{bmatrix},

where:
c_{i,j} = \sum_{r=1}^n a_{i,r}b_{r,j} \;\;\; \left(i=1, 2, \ldots m;\;j=1, 2, \ldots q \right).

Therefore, in order for matrix multiplication to be defined, the dimensions of the matrices must satisfy
(n \times m)(m \times p)=(n \times p)

Note that matrix multiplication is not commutative (unless A and B are diagonal and of the same dimension).

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