Differences between revisions 5 and 7 (spanning 2 versions)
⇤ ← Revision 5 as of 2022-03-19 20:45:13 →
Size: 1268
Editor: DominicRicottone
Comment:
← Revision 7 as of 2024-01-21 17:51:51 → ⇥
Size: 2092
Editor: DominicRicottone
Comment: Expanded on permutations, removed properties
Deletions are marked like this. Additions are marked like this.
Line 2: Line 2:

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

<<TableOfContents>>

----
Line 5: Line 13:
The '''identity matrix''' multiplied by matrix A returns matrix A. The '''identity matrix''' is a diagonal line of 1s in a matrix of 0s.
Line 7: Line 15:
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.
Line 10: Line 18:
┌ ┐
│ 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
Line 16: Line 26:

----
Line 21: Line 33:
A '''permutation matrix''' multiplied by matrix A returns 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.
Line 24: Line 36:
┌ ┐┌ ┐ ┌ ┐
│ 0 1││ 1 2│ │ 3 4│
│ 1 0││ 3 4│=│ 1 2│
└ ┘└ ┘ └ ┘		 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
Line 30: Line 61:
See [[LinearAlgebra/PermutationMatrices|Permutation Matrices]] for more information. 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
Line 34: Line 67:
== Inverse Matrices == === Counting Permutations ===
Line 36: Line 69:
An '''inverse matrix''' A^-1^ multiplied by matrix A returns the identity matrix.

If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible.

See [[LinearAlgebra/MatrixInversion|Matrix Inversion]] for more information.



== Symmetric Matrices ==

A '''symmetric matrix''' is any matrix that is equal to its [[LinearAlgebra/MatrixTransposition|transpose]].		 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,,.
Line 49: Line 72:
┌ ┐
│ 1 7│
│ 7 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
Line 55: Line 81:
See [[LinearAlgebra/MatrixTransposition#SymmetricMatrices|Symmetric Matrices]] for more information. 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.

CategoryRicottone

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