_03e_multiply Namespace Reference

Matrix multiplication: \(A \times B\). More...

Functions
def	dotProd (a, b)
	Dot product of two vectors. More...
def	transpose (m, pythonicWay=False)
	Transpose a given matrix. More...
def	matMultiply (m1, m2)
	Multiplies two matrices. More...
def	matMultiply2 (m1, m2)
	Multiplies two matrices. More...
def	matMultiply3 (m1, m2)
	Multiplies two matrices. More...
def	main ()

Detailed Description

Matrix multiplication: \(A \times B\).

\[\mathbf{A}=\begin{pmatrix} A_{11} & A_{12} & \cdots & A_{1m} \\ A_{21} & A_{22} & \cdots & A_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \cdots & A_{nm} \\ \end{pmatrix}, \quad\mathbf{B}=\begin{pmatrix} B_{11} & B_{12} & \cdots & B_{1p} \\ B_{21} & B_{22} & \cdots & B_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ B_{m1} & B_{m2} & \cdots & B_{mp} \\ \end{pmatrix} \]

Given two matrices A and B, (where, necessarily, the number of columns in A equals the number of rows in B, which equals m) the matrix product AB is defined by:

\[\mathbf{A}\mathbf{B} =\begin{pmatrix} (AB)_{11} & (AB)_{12} & \cdots & (AB)_{1p} \\ (AB)_{21} & (AB)_{22} & \cdots & (AB)_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ (AB)_{n1} & (AB)_{n2} & \cdots & (AB)_{np} \\ \end{pmatrix} \]

(with no multiplication signs or dots) where AB has entries defined by:

\[(AB)_{ij} = \sum_{k=1}^m A_{ik}B_{kj}.\]

Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also their vector dot product:

\[\mathbf{a}_i=\begin{pmatrix} A_{i1} & A_{i2} & \cdots & A_{im} \end{pmatrix}\,, \quad \mathbf{b}_j=\begin{pmatrix} B_{1j} \\ B_{2j} \\ \vdots \\ B_{mj} \end{pmatrix}, \quad (AB)_{ij} = \mathbf{a}_i \cdot \mathbf{b}_j. \]

Author

Paulo Roma

Since

16/04/2009

See also

http://en.wikipedia.org/wiki/Matrix_multiplication

Function Documentation

dotProd()

def _03e_multiply.dotProd	(		a,
			b
	)

Dot product of two vectors.

Parameters

a	first vector.
b	second vector.

Returns

a . b

Referenced by _05c_cnpj.areValidDigits(), _05b_cpf.areValidDigits(), and matMultiply().

main()

def _03e_multiply.main

(

)

References matMultiply(), matMultiply2(), and matMultiply3().

matMultiply()

def _03e_multiply.matMultiply	(		m1,
			m2
	)

Multiplies two matrices.

First solution, "C" or Pascal like.

Parameters

m1	first matrix.
m2	second matrix.

Returns

m1 x m2.

References dotProd(), and transpose().

Referenced by main().

matMultiply2()

def _03e_multiply.matMultiply2	(		m1,
			m2
	)

Multiplies two matrices.

Second solution, using list comprehension.

Parameters

m1	first matrix.
m2	second matrix.

Returns

m1 x m2.

Referenced by main().

matMultiply3()

def _03e_multiply.matMultiply3	(		m1,
			m2
	)

Multiplies two matrices.

Third solution, python like, using zip.

Parameters

m1	first matrix.
m2	second matrix.

Returns

m1 x m2.

See also

https://codeyarns.com/2012/04/26/unpack-operator-in-python/

http://hangar.runway7.net/python/packing-unpacking-arguments

https://docs.python.org/3/library/functions.html#zip

Referenced by main().

transpose()

def _03e_multiply.transpose	(		m,
			pythonicWay = False
	)

Transpose a given matrix.

Parameters

m	matrix.
pythonicWay	selects the algorithm to use.

Returns

\(m^T = zip(*m)\)

Referenced by matMultiply().