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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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| Given an [[LinearAlgebra/SpecialMatrices#Upper_Triangular_Matrices|upper triangular matrix]], the determinant is the product of the diagonal. | The determinant of '''''A''''' is notated as ''|'''A'''|''. |
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| 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. | The determinant of any non-square matrix is 0. |
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| As noted in the properties below, elimination does not change the determinant (although row exchanges flip the sign of it). | Given a matrix of shape 2 by 2, the determinant is calculated like: |
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| Given a [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]], the determinant is the product of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. | {{{ | a b | det | c d | = ad - bc }}} |
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| 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| └ ┘
└ ┘