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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 ones in a matrix of zeros.
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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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A '''permutation matrix''' multiplied by matrix A returns matrix C which is a row-exchanged transformation of A. 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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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,,. {{{
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: '''''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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Note that a permutation matrix can mirror either the rows or columns of matrix A, depending simply on the order. For any ''n'' by ''n'' matrix, there are ''n''! possible permutation matrices.
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{{{
┌ ┐┌ ┐ ┌ ┐
│ 0 1││ 1 2│ │ 3 4│
│ 1 0││ 3 4│=│ 1 2│
└ ┘└ ┘ └ ┘

┌ ┐┌ ┐ ┌ ┐
│ 1 2││ 0 1│ │ 2 1│
| 3 4|│ 1 0│=│ 4 3│
└ ┘└ ┘ └ ┘
}}}
----
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== Inverse Matrices == == Upper Triangular Matrices ==
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An '''inverse matrix''' is denoted as A^-1^. If a matrix is multiplied by its inverse matrix, it returns the identity matrix. If a square matrix has only zeros below the diagonal, it is an '''upper triangular matrix'''.
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If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] results in a '''row echelon form''' of '''''A''''' which, if '''''A''''' is square, is also upper triangular.
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For a square matrix A, the left inverse is the same as the right inverse. AA^-1^ = A^-1^A = I [[LinearAlgebra/Orthonormalization|Gram-Schmidt orthonormalization]] is characterized as '''''A = QR''''' where '''''R''''' is upper triangular.

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== Lower Triangular Matrices ==

If a square matrix has only zeros above the diagonal, it is a '''lower triangular matrix'''.

If [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] is continued into backwards elimination, it results in a '''reduced row echelon form''' of '''''A'''''. If '''''A''''' is square, it will be both upper and lower triangular.

Special Matrices

These special matrices are core concepts to linear algebra.


Identity Matrix

The identity matrix is a diagonal line of ones in a matrix of zeros.

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.


Upper Triangular Matrices

If a square matrix has only zeros below the diagonal, it is an upper triangular matrix.

Gauss-Jordan elimination results in a row echelon form of A which, if A is square, is also upper triangular.

Gram-Schmidt orthonormalization is characterized as A = QR where R is upper triangular.


Lower Triangular Matrices

If a square matrix has only zeros above the diagonal, it is a lower triangular matrix.

If Gauss-Jordan elimination is continued into backwards elimination, it results in a reduced row echelon form of A. If A is square, it will be both upper and lower triangular.


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