Determinants

The determinant is a number that embeds most information about a matrix, much like the trace. Most importantly it is the scaling factor in space for a linear transformation.

Definition

The determinant is the product of eigenvalues: Π_i λ_i.

There is an important connection between determinants and elimination. A matrix does not need to be eliminated to arrive at the determinant, but if a matrix cannot be eliminated into an upper triangular matrix, it is singular and degenerate and non-invertible and the determinant is 0. This generally only happens if there is multicolinearity. This does lead to a convenient test for invertibility.

The determinant of A is notated as |A|.

Simple Case

Given a matrix of shape 2 x 2, the determinant is calculated like:

    | a b |
det | c d | = ad - bc

Properties

The determinant of any non-square matrix is 0.

Determinants can be factored: |AB| = |A| |B|.

The determinant of the inverse is the inverse of the determinant: |A^-1| = 1/|A|.

Transposition does not change the determinant: |A^T| = |A|.

Special Matrices

The determinant of the identity matrix is 1.

The determinant of a permutation matrix is 1 or -1; 1 if there are an even number of row exchanges; and -1 if there are an odd number.

The determinant of an orthogonal matrix is 1 or -1.

Large, sparse matrices can be broken up.

    ┌        ┐
    | 2 0 0 0|           ┌    ┐
    | 0 a b 0|           │ a b│
det | 0 c d 0| = 2 * det │ c d│ * 3
    | 0 0 0 3|           └    ┘
    └        ┘

Elimination

Elimination does not necessarily change the determinant. More specifically, elimination of A into U is often characterized as U = EA; left multiplication by one or more elimination matrices. It follows from the above properties that |U| = |E| |A|. If the determinant of E is 1, then clearly elimination does not change the determinant.

Adding (or subtracting) a linear combination of one row to another does not change the determinant. The example here featured only transformations like this ("subtracting a multiple of the pivot row from the targeted row"), and it can be shown that the determinant is unchanged.

julia> using LinearAlgebra

julia> A = [1 2 1; 3 8 1; 0 4 1]
3×3 Matrix{Int64}:
 1  2  1
 3  8  1
 0  4  1

julia> det(A)
10.0

julia> B = [1 2 1; 0 2 -2; 0 4 1]
3×3 Matrix{Int64}:
 1  2   1
 0  2  -2
 0  4   1

julia> det(B)
10.0

To prove this, consider the following:

    ┌          ┐       ┌    ┐       ┌      ┐       ┌    ┐           ┌    ┐       ┌    ┐
    │    a    b│       │ a b│       │  a  b│       │ a b│           │ a b│       │ a b│ 
det │ c-ma d-mb│ = det │ c d│ - det │ ma mb│ = det │ c d│ - m * det │ a b│ = det │ c d│ - m * 0
    └          ┘       └    ┘       └      ┘       └    ┘           └    ┘       └    ┘

More generally, adding (or subtracting) to one row of a matrix changes the determinant in a manner that can look like 'factoring out' the addition.

    ┌        ┐       ┌    ┐       ┌    ┐
    │ a+x b+y│       │ a b│       │ x y│
det │   c   d│ = det │ c d│ + det │ c d│
    └        ┘       └    ┘       └    ┘

Multiplying one row of a matrix by some scalar multiplies the determinant by the same scalar. Or more flexibly, multiplying n rows by some scalar t also multiplies the determinant by tⁿ.

    ┌      ┐           ┌    ┐
    │ ta tb│           │ a b│
det │  c  d│ = t * det │ c d│
    └      ┘           └    ┘

    ┌      ┐           ┌      ┐               ┌    ┐
    │ ta tb│           │  a  b│               │ a b│
det │ tc td│ = t * det │ tc td│ = t * t * det | c d|
    └      ┘           └      ┘               └    ┘

A row exchange is characterized by a permutation matrix with a determinant of -1. Therefore the determinant's sign is flipped for every row exchange.

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