Special Matrices

These special matrices are core concepts to linear algebra.

Identity Matrix

The identity matrix is a diagonal line of ones in a square 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 is a square matrix of zeros with a one in each row. Multiplying a permutation matrix by some matrix A (PA) results in a row-exchanged A. Multiplying some matrix A by a permutation matrix (AP) results in a column-exhanged A.

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^TP = 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}.

┌      ┐          ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐ ┌      ┐
│ 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.

Symmetric Matrices

A symmetric matrix is equal to its transpose.

Only a square matrix can be symmetric.

These matrices have several useful properties:

• The eigenvalues are always real.

• The eigenvectors can be chosen to be orthonormal.

• The diagonalization of a symmetric matrix is expressed as A = QΛQ^-1 = QΛQ^T using the orthonormal eigenvectors.

• The signs of the pivots are the same as the signs of the eigenvalues.

Projection matrices are always symmetric, and symmetric matrices are combinations of orthogonal projection matrices.

Multiplying a rectangular matrix R by its transpose R^T will always create a symmetric matrix. This can be proven with the above properties: (R^TR)^T = R^T(R^T)^T = R^TR.

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