Matrix Arithmetic
Matrix Introduction
Matrix is a two-dimensional array of numbers often denoted with uppercase letter. Matrix is comprised of rows(m) and columns(n).
Vector may be considered a matrix with one column and multiple rows.
Create a matrix using NumPy
Matrix Addition
Two matrices with the same dimensions can be added to create a new third matrix. C = A + B
Matrix-Scalar Multiplication
A matrix can be multiplied by a scalar represented asC = A . b
where b is a scalar. Note: use '*' in Numpy for matrix multiplication
Matrix-Vector Multiplication
A matrix can be multiplied by a vector represented as C = A . v
where v is a vector given it follows the rule of matrix multiplication.
Rule of matrix multiplication
For example, matrix A has the dimensions m rows and n columns and matrix B has the dimensions n and k. The n columns in A and n rows b are equal. The result is a new matrix with m rows and k columns.
Example of Matrix-Vector multiplication:
Matrix Multiplication (Hadamard Product)
Two matrices with the same dimension can be simply multiplied element-wise. Also known as Hadamard Product, it is represented with a circle as C = A o B
Note: Implemented using star operator as in Matrix-Scalar Multiplication
Matrix-Matrix Multiplication (Dot Product)
Matrix-Matrix, also called the matrix dot product is a complicated multiplication which must follow the rule of matrix multiplication represented as C = A * B
.
This is made clear with the following image:
A: 4 x 2; B: 2 x 3; C = A * B => 4 x 3 For calculating: C12(marked with red arrow) = a11*b12 + a12*b22 C33(marked with blue arrow) = a31*b13 + a32*b23
Link: Introduction to Matrices and Matrix Arithmetic for Machine Learning
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