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These '''special matrices''' are core concepts to linear algebra.

<<TableOfContents>>

----
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The '''identity matrix''' multiplied by matrix A returns matrix A. The '''identity matrix''' is a diagonal line of 1s in a matrix of 0s.
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This matrix is simply a diagonal line of 1s in a matrix of 0s. Any matrix A multiplied by the (appropriately sized) identity matrix returns matrix A.
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┌ ┐
│ 1 0 0│
│ 0 1 0│
│ 0 0 1│
└ ┘		 julia> using LinearAlgebra

julia> Matrix{Int8}(I,3,3)
3×3 Matrix{Int8}:
 1 0 0
 0 1 0
 0 0 1
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----
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== Permutation Matrix == == Permutation Matrices ==
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The '''permutation matrix''' multiplied by matrix A returns matrix C which is a mirrored transformation of A.

This matrix is simply a diagonal line of 1s in a matrix of 0s, but going the opposite direction as compared to an identity matrix		 A '''permutation matrix''' multiplied by matrix A returns a row- or column-exchanged transformation of A. If the permutation matrix leads in the multiplication, rows are exchanged. If the permutation matrix follows, columns are exchanged.
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┌ ┐
│ 0 0 1│
│ 0 1 0│
│ 1 0 0│
└ ┘		 julia> P = Matrix{Int8}(I,3,3)[:,[3,2,1]]
3×3 Matrix{Int8}:
 0 0 1
 0 1 0
 1 0 0

julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Matrix{Int64}:
 1 2 3
 4 5 6
 7 8 9

julia> P * A
3×3 Matrix{Int64}:
 7 8 9
 4 5 6
 1 2 3

julia> A * P
3×3 Matrix{Int64}:
 3 2 1
 6 5 4
 9 8 7
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Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order. The transpose permutation matrix is the same as the inverse permutation matrix: P^T^ = P^-1^.

The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: P^T^P = I



=== Counting Permutations ===

For 3 by 3 matrices, there are 6 possible permutation matrices. They are often denoted based on the rows they exchange, such as P,,2 3,,.
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┌ ┐┌ ┐ ┌ ┐
│ 0 1│| 1 2| | 3 4|
│ 1 0│| 3 4|=| 1 2|
└ ┘└ ┘ └ ┘		 ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ 1 0 0│ │ 1 0 0│ │ 0 1 0│ │ 0 1 0│ │ 0 0 1│ │ 0 0 1│
│ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│
│ 0 0 1│ │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 1 0 0│
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘
(identity matrix) P P (and so on...)
                     2,3 1,2
}}}
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┌ ┐┌ ┐ ┌ ┐
| 1 2|│ 0 1│ | 2 1|
| 3 4|│ 1 0│=| 4 3|
└ ┘└ ┘ └ ┘
}}}		 For any ''n'' by ''n'' matrix, there are ''n''! possible permutation matrices.

Special Matrices

These special matrices are core concepts to linear algebra.

Contents

  1. Special Matrices
    1. Identity Matrix
    2. Permutation Matrices
      1. Counting Permutations

Identity Matrix

The identity matrix is a diagonal line of 1s in a matrix of 0s.

Any matrix A multiplied by the (appropriately sized) identity matrix returns matrix A. 

julia> using LinearAlgebra

julia> Matrix{Int8}(I,3,3)
3×3 Matrix{Int8}:
 1  0  0
 0  1  0
 0  0  1

Permutation Matrices

A permutation matrix multiplied by matrix A returns a row- or column-exchanged transformation of A. If the permutation matrix leads in the multiplication, rows are exchanged. If the permutation matrix follows, columns are exchanged. 

julia> P = Matrix{Int8}(I,3,3)[:,[3,2,1]]
3×3 Matrix{Int8}:
 0  0  1
 0  1  0
 1  0  0

julia> A = [1 2 3; 4 5 6; 7 8 9]
3×3 Matrix{Int64}:
 1  2  3
 4  5  6
 7  8  9

julia> P * A
3×3 Matrix{Int64}:
 7  8  9
 4  5  6
 1  2  3

julia> A * P
3×3 Matrix{Int64}:
 3  2  1
 6  5  4
 9  8  7

The transpose permutation matrix is the same as the inverse permutation matrix: PT = P-1.

The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: PTP = I

Counting Permutations

For 3 by 3 matrices, there are 6 possible permutation matrices. They are often denoted based on the rows they exchange, such as P2 3. 

┌      ┐          ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐
│ 1 0 0│          │ 1 0 0│ │ 0 1 0│ │ 0 1 0│ │ 0 0 1│ │ 0 0 1│
│ 0 1 0│          │ 0 0 1│ │ 1 0 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│
│ 0 0 1│          │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 1 0 0│
└      ┘          └      ┘ └      ┘ └      ┘ └      ┘ └      ┘
(identity matrix)   P        P        (and so on...)
                     2,3      1,2

For any n by n matrix, there are n! possible permutation matrices.

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LinearAlgebra/SpecialMatrices (last edited 2024-01-30 15:45:39 by DominicRicottone)