Multiplication of matrices

Definition

For general matrices A and B

multiplication of matrices (or inner/dot-product of row and column) is defined if and only if n₁ = m₂ = u.

Corollary: A⋅B and B⋅A are defined if and only if n₁ = m₂ = u and n₂ = m₁ = v.

Note that the dot-product of row and column can be considered the multiplication of the i row and j column of respective matrices. Therefore, dot-product is also called inner product.

mtimes operator

Import the mtimes operator in the namespace by doing

=> (require '[kigubkur.op [mtimes :refer [mtimes]]])

Examples using mtimes

Example 1

=> (def A [[1]
           [2]
           [3]])
=> (def B [[2 3 4]])
=> (mtimes A B)
[[2 3 4]
 [4 6 8]
 [6 9 12]]

Example 2

=> (def A [[2 5]
           [8 10]])
=> (def B [[5 4]
           [50 40]])
=> (mtimes A B)
[[260 208]
 [540 432]]

Example 3

=> (def A [[1 -1 2]
           [0 3 4]])
=> (def B [[2 7]
           [-1 1]
           [5 -4]])
=> (mtimes A B)
[[13 -2]
 [17 -13]]

Example 4

=> (def A [[6 9]
           [2 3]])
=> (def B [[2 6 0]
           [7 9 8]])
=> (mtimes A B)
[[75 117 72]
 [25 39 24]]

Commutativity of multiplication of matrices

Assuming AB and BA are defined, in general AB ≠ BA.

=> (def A [[1 -2 3]
           [-4 2 5]])
=> (def B [[2 3]
           [4 5]
           [2 1]])
=> (mtimes A B)
[[0 -4]
 [10 3]]
=> (mtimes B A)
[[-10 2 21]
 [-16 2 37]
 [-2 -2 11]]

Special case of commutativity

If A and B are diagonal matrices of same order then AB = BA.

=> (def A [[1 0]
           [0 2]])
=> (def B [[3 0]
           [0 4]])
=> (mtimes A B)
[[3 0]
 [0 8]]
=> (= (mtimes A B) (mtimes B A))
true

Product of two non-zero matrices can be a zero matrix

Matrices A, B or both does not have to be a zero matrix for their product to be a zero matrix.

=> (def A [[0 -1]
           [0 2]])
=> (def B [[3 5]
           [0 0]])
=> (mtimes A B)
[[0 0]
 [0 0]]

Basic properties of matrix multiplication

1. Associative Law

=> (def A [[1 1 -1]
           [2 0 3]
           [3 -1 2]])
=> (def B [[1 3]
           [0 2]
           [-1 4]])
=> (def C [[1 2 3 -4]
           [2 0 -2 1]])
=> (= (mtimes (mtimes A B) C) (mtimes A (mtimes B C)))
true

2. Distributive Law

=> (require '[kigubkur.op [plus :refer [plus]]])

2.1. Given that the + and ⋅ operations on the matrices A, B and C are defined

=> (def A [[1 1]
           [-1 3]])
=> (def B [[1 2]
           [-4 2]])
=> (def C [[-7 6]
           [3 2]])
=> (= (mtimes A (plus B C)) (plus (mtimes A B) (mtimes A C)))
true

2.2. Given that the + and ⋅ operations on the matrices A, B and C are defined

=> (def A [[0 6 7]
           [-6 0 8]
           [7 -8 0]])
=> (def B [[0 1 1]
           [1 0 2]
           [1 2 0]])
=> (def C [[2]
           [-2]
           [3]])
=> (= (mtimes (plus A B) C) (plus (mtimes A C) (mtimes B C)))
true

3. Existence of multiplicative identity

=> (def A [[2 0 1]
           [2 1 3]
           [1 -1 0]])
=> (def I [[1 0 0]
           [0 1 0]
           [0 0 1]])
=> (= (mtimes I A) (mtimes A I) A)
true