Determinants
The determinant is a number that embeds most information about a square matrix. Chiefly, for a matrix used to transform bases, the determinant is the scaling factor of space in the transformation.
Contents
Definition
The determinant of A is notated as |A|.
The determinant of any non-square matrix is 0.
Given a matrix of shape 2 by 2, the determinant is calculated like:
| a b | det | c d | = ad - bc
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 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 invertability.
Properties
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 determinent: |AT| = |A|.
Exchanging rows flips the sign of the determinant. This is the intuitive explanation for the determinant of the permutation matrix as noted above. If U = PA, then |U| = |P| |A|.
Multiplying a single row of a matrix by some factor simply means that the determinant was multiplied by the same factor.
┌ ┐ ┌ ┐
│ ta tb│ │ a b│
det │ c d│ = t * det │ c d│
└ ┘ └ ┘Multiplying every row of a matrix by some factor means that the determinant was multiplied by the same factor to the nth power.
┌ ┐ ┌ ┐ ┌ ┐
│ ta tb│ │ a b│ │ a b│
det │ tc td│ = t * det │ tc td│ = t * t * det | c d|
└ ┘ └ ┘ └ ┘Adding to or subtracting from a single row of a matrix means that the determinant is the sum of the determinants of the two factored-out matrices.
┌ ┐ ┌ ┐ ┌ ┐
│ a+x b+y│ │ a b│ │ x y│
det │ c d│ = det │ c d│ + det │ c d│
└ ┘ └ ┘ └ ┘Furthermore, elimination does not change the determinant at all.
┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐
│ 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
└ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘
Special Matrices
For the identity matrix, the determinant is 1.
For a permutation matrix, the determinant is 1 if there are an even number of row exchanges in the matrix or -1 if there are an odd number of row exchanges.
For an orthogonal matrix, the determinant is 1 or -1.
For an upper triangular matrix, the determinant is the product of the diagonal.
For a diagonal matrix, the determinant is the product of the eigenvalues. If a matrix cannot be diagonalized, it is defective and one of the eigenvalues is zero. It may still be invertible.
Large matrices, especially with mostly zeros, 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| └ ┘
└ ┘