Special Matrices

These special matrices are core concepts to linear algebra.

Contents

  1. Special Matrices
    1. Identity Matrix
    2. Permutation Matrices
      1. Counting Permutations
    3. Upper Triangular Matrices
    4. Lower Triangular Matrices

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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