= Matrix Multiplication =

'''Matrix multiplication''' is a fundamental operation.

<<TableOfContents>>

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== Dimensions ==

To multiply '''a matrix by another matrix''' as '''''AB''' = '''C''''', they must have a common dimension. '''''A''''' must be as wide as '''''B''''' is tall, and the product will be as wide as '''''B''''' and as tall as '''''A'''''. Alternatively: '''''A''''' has shape ''m'' rows by ''n'' columns, and '''''B''''' has shape ''n'' rows by ''p'' columns, so the product '''''C''''' will have ''m'' rows and ''p'' columns.

To multiply '''a matrix by a vector''' as '''''A'''x = y'', the vector can be seen as a matrix with ''n'' rows and 1 column. The product will also have 1 column, i.e. be a vector.

To multiply '''a vector by a matrix''', the vector must be [[LinearAlgebra/MatrixTransposition|transposed]] so that it has ''n'' columns and 1 row. In other words, the multiplication is as ''x^T^'''A''' = y^T^''. Alternatively, the multiplication is as ''('''A'''^T^x)^T^ = y^T^''.

For multiplying vectors, see [[LinearAlgebra/VectorMultiplication|vector multiplication]].

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== Properties ==

Matrix multiplication is taking linear combinations of the rows of '''''A''''' according to the columns of '''''B''''', or vice versa.

Matrix multiplication is not commutative.

{{{
julia> A = [1 2; 0 0; 0 0]
3×2 Matrix{Int64}:
 1  2
 0  0
 0  0

julia> B = [1 0 0; 2 0 0]
2×3 Matrix{Int64}:
 1  0  0
 2  0  0

julia> A * B
3×3 Matrix{Int64}:
 5  0  0
 0  0  0
 0  0  0

julia> B * A
2×2 Matrix{Int64}:
 1  2
 2  4
}}}

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== Cell-wise Computation ==

A cell in a matrix is expressed as '''''A''',,ij,,'' where ''i'' is a row index and ''j'' is a column index. Indexing starts at 1.

For '''''C''' = '''AB''''': '''''C''',,ij,,'' can be computed as the [[LinearAlgebra/VectorMultiplication#Dot_Product|dot product]] of row '''''A''',,i,,'' and column '''''B''',,j,,''.

Referencing the complete solution above:

{{{
julia> A[1, :]
2-element Vector{Int64}:
 1
 2

julia> B[:, 1]
2-element Vector{Int64}:
 1
 2

julia> using LinearAlgebra

julia> dot(A[1, :], B[:, 1])
5
}}}

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== Column-wise Computation ==

Column '''''C''',,j,,'' is a linear combination of all columns in '''''A''''' taken according to the column '''''B''',,j,,''.

Referencing the complete solution above and recall that '''''B''',,1,, = [1 2]'':

{{{
C  = 1*A  + 2*A
 1      1      2

C  = 1*[1 0 0] + 2*[2 0 0]
 1

C  = [1 0 0] + [4 0 0]
 1

C  = [5 0 0]
 1
}}}

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== Row-wise Computation ==

Row '''''C''',,i,,'' is a linear combination of all rows in '''''B''''' taken according to the row '''''A''',,i,,''.

Referencing the complete solution above and recall that '''''A''',,1,, = [1 2]'':

{{{
C  = 1*B  + 2*B
 1      1      2

C  = 1*[1 0 0] + 2*[2 0 0]
 1

C  = [1 0 0] + [4 0 0]
 1

C  = [5 0 0]
 1
}}}

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== Block-wise Computation ==

Matrix multiplication can be evaluated in blocks. Suppose '''''A''''' and '''''B''''' are 20x20 matrices; they can be divided each into 10x10 quadrants.

Using these matrices '''''A''''' and '''''B''''' as the building blocks:

{{{
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> B = [1 0; 0 1]
2×2 Matrix{Int64}:
 1  0
 0  1
}}}

Much larger matrices may be composed of these building blocks.

{{{
julia> A_ = [A1 A2; A3 A4]
4×4 Matrix{Int64}:
 1  2  1  2
 3  4  3  4
 1  2  1  2
 3  4  3  4

julia> B_ = [B1 B2; B3 B4]
4×4 Matrix{Int64}:
 1  0  1  0
 0  1  0  1
 1  0  1  0
 0  1  0  1
}}}

The entire product could be computed:

{{{
julia> A_ * B_
4×4 Matrix{Int64}:
 2  4  2  4
 6  8  6  8
 2  4  2  4
 6  8  6  8
}}}

But if a specific block of the product is of interest, it can be solved like '''''C'''^1^ = '''A'''^1^'''B'''^1^ + '''A'''^2^'''B'''^3^''.

{{{
julia> A1 * B1 + A2 * B3
2×2 Matrix{Int64}:
 2  4
 6  8
}}}



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