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| The '''determinant''' is a number that embeds most information about a matrix. For a matrix effecting a [[LinearAlgebra/Basis#Change_of_Basis|change of basis]], the determinant is the scaling factor of space between bases. The determinant of '''''A''''' is often notated as ''|'''A'''|''. |
The '''determinant''' is a number that embeds most information about a square matrix. Chiefly, for a matrix used to transform [[LinearAlgebra/Basis#Change_of_Basis|bases]], the determinant is the scaling factor of space in the transformation. |
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| The determinant of '''''A''''' is notated as ''|'''A'''|''. The determinant of any non-square matrix is 0. |
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| Given an [[LinearAlgebra/SpecialMatrices#Upper_Triangular_Matrices|upper triangular matrix]], the determinant is the product of the diagonal. If a matrix cannot be [[LinearAlgebra/Elimination|eliminated]] into an upper triangular matrix, it is '''degenerate''' and [[LinearAlgebra/MatrixProperties#Invertible|non-invertible]]. This directly means that the determinant is zero. This generally only happens if there is multicolinearity. As noted in the properties below, elimination does not change the determinant (although row exchanges flip the sign of it). Given a [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]], the determinant is the product of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. If a matrix cannot be [[LinearAlgebra/Diagonalization|diagonalized]], it is '''defective'''. This simply means that one of the eigenvalues is zero; it may still be invertible and therefore have a non-zero determinent. ---- |
There is an important connection between determinants and [[LinearAlgebra/Elimination|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 [[LinearAlgebra/Invertability|invertability]]. |
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| == Test for Invertibility == | |
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| There is a direct connection between [[LinearAlgebra/MatrixProperties#Invertible|invertibility]] and having a non-zero determinant. As a result, calculation of the determinant is a simple test for invertibility. ---- == Special Matrices and their Determinants == For the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]], the determinant is 1. For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|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 [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrix]], the determinant is 1 or -1. ---- == Properties == |
=== Properties === |
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| The determinant of the [[LinearAlgebra/MatrixInversion|inverse]] is the inverse of the determinant: ''|'''A'''^-1^| = 1/|'''A'''|''. | The determinant of the [[LinearAlgebra/Invertability|inverse]] is the inverse of the determinant: ''|'''A'''^-1^| = 1/|'''A'''|''. |
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| [[LinearAlgebra/MatrixTransposition|Transposition]] does not change the determinent: ''|'''A'''^T^| = |'''A'''|''. | [[LinearAlgebra/Transposition|Transposition]] does not change the determinent: ''|'''A'''^T^| = |'''A'''|''. |
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| ---- == Special Matrices == For the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]], the determinant is 1. For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|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 [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrix]], the determinant is 1 or -1. For an [[LinearAlgebra/SpecialMatrices#Upper_Triangular_Matrices|upper triangular matrix]], the determinant is the product of the diagonal. For a [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]], the determinant is the product of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. If a matrix cannot be [[LinearAlgebra/Diagonalization|diagonalized]], it is '''defective''' and one of the eigenvalues is zero. It may still be invertible. |
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| └ ┘
└ ┘