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| = Matrix Inversion = | = Invertibility = |
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| For some matrices '''''A''''', the '''inverse matrix''' ('''''A'''^-1^'') is a matrix which can be multiplied by the original matrix to produce the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. The calculation of an inverse matrix, if it exists, is called '''inversion'''. | '''Invertibility''' is a property of square matrices. If a matrix is invertible, there is an inverse matrix that it can be multiplied by to produce the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. The calculation of an inverse matrix is '''inversion'''. |
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| An inverse matrix satisfies the equation '''''AA'''^-1^ = '''I'''''. | A matrix '''''A''''' is invertible if there is a matrix '''''A'''^-1^'' which satisfies '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I'''''. |
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| Not all matrices have an inverse that can satisfy that condition. If '''''A'''^-1^'' exists, then '''''A''''' is '''invertible''' and '''non-singular'''. | If a matrix cannot be inverted, it is '''singular''' and '''degenerate''' and '''non-invertible'''. Only square matrices can be invertible. However, a non-square matrix can separably have distinct '''left inverse''' and '''right inverse''' matrices. Generally, if ''m < n'', then a matrix with shape ''m'' by ''n'' and rank of ''m'' can have a right inverse; a matrix with shape ''n'' by ''m'' and rank of ''m'' can have a left inverse. |
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| The core principle of inversions is that a matrix '''''A''''' can be canceled out from a larger equation. '''''AA'''^-1^ = '''I''''', so the two terms cancel out. | By definition, '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I'''''. |
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| An invertible matrix has only one vector in the [[LinearAlgebra/NullSpaces|null space]]: the zero vector. | An invertible matrix is square and is full [[LinearAlgebra/Rank|rank]]. |
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| For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrix]] '''''P''''', the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: '''''P'''^-1^ = '''P'''^T^''. | An invertible matrix has only one vector in the [[LinearAlgebra/NullSpace|null space]]: the zero vector. If any basis vector of a matrix is a linear transformation of another, then the matrix does not have [[LinearAlgebra/Basis|basis]] and must be non-invertible. |
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| For a square matrix '''''A''''', the '''left inverse''' is the same as the '''right inverse'''. '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I''''' | For [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the inverse is also the [[LinearAlgebra/Transposition|transpose]]: '''''Q'''^-1^ = '''Q'''^T^''. |
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| == Calculation == | == Test with Determinant == |
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| Consider the below system, which shows an unknown matrix ('''''A'''^-1^'') multiplied by a known matrix ('''''A''''') creating an [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] ('''''I'''''). | The [[LinearAlgebra/Determinant|determinant]] is the most common test for invertibility. If ''|'''A'''| != 0'', then '''''A''''' is invertible. If ''|'''A'''| = 0'', then '''''A''''' is non-invertible. ---- == Calculation with Determinant == Consider the ''2 x 2'' case: |
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| -1 A A = I ┌ ┐┌ ┐ ┌ ┐ │ 1 3││ a b│ │ 1 0│ │ 2 7││ c d│ = │ 0 1│ └ ┘└ ┘ └ ┘ |
┌ ┐ │ a b │ │ c d │ └ ┘ |
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| The inverse matrix is calculated with [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reverse elimination]]. [[LinearAlgebra/Elimination#Simplification_with_Augmented_Matrices|Augment]] '''''A''''' with '''''I'''''. | The inverse matrix is |
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| The elimination proceeds as: | {{{ 1 ┌ ┐ ―――――― │ d -b │ Det(A) │ -c a │ └ ┘ }}} ---- == Calculation with Elimination == Because '''''AA'''^-1^ = '''I''''', applying [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|backwards elimination]] on '''''A''''' augmented with an [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] ('''''I''''') will create '''''A'''^-1^'' in the augmentation. |
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| }}} The reverse elimination proceeds as: {{{ |
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| ---- == Calculation with Determinants and Cofactor Matrices == Given the [[LinearAlgebra/Determinant|determinant]] of '''''A''''', it can also be simple to compute '''''A'''^-1^'' as ''(1/|'''A'''|)'''C'''^T^''. '''''C''''' is the cofactor matrix, where ''c,,i j,,'' is the cofactor of ''a,,i j,,''. For example, given a 2 x 2 '''''A''''' like: {{{ ┌ ┐ │ a b│ │ c d│ └ ┘ }}} The cofactor matrix '''''C''''' is: {{{ ┌ ┐ │ d -c│ │ -b a│ └ ┘ }}} But this must be transposed to '''''C'''^T^'': {{{ ┌ ┐ │ d -b│ │ -c a│ └ ┘ }}} And then '''''A'''^-1'' is: {{{ ┌ ┐ │ (1/det A) * d (1/det A) * -b│ │ (1/det A) * -c (1/det A) * a│ └ ┘ }}} The above example fits into this formula. The [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|backwards elimination]] prove that the determinant of that '''''A''''' is 1. The more fundamental formula ''ad - bc'' expands to ''1 * 7 - 2 * 3'' which also reveals a determinant of 1. As such, ''(1/|'''A'''|)'' is trivially 1. So simply plug the given (''a'', ''b'', ''c'', ''d'') into the transposed cofactor matrix to find the inverse. |
Invertibility
Invertibility is a property of square matrices. If a matrix is invertible, there is an inverse matrix that it can be multiplied by to produce the identity matrix. The calculation of an inverse matrix is inversion.
Contents
Definition
A matrix A is invertible if there is a matrix A-1 which satisfies AA-1 = A-1A = I.
If a matrix cannot be inverted, it is singular and degenerate and non-invertible.
Only square matrices can be invertible. However, a non-square matrix can separably have distinct left inverse and right inverse matrices. Generally, if m < n, then a matrix with shape m by n and rank of m can have a right inverse; a matrix with shape n by m and rank of m can have a left inverse.
Properties
By definition, AA-1 = A-1A = I.
An invertible matrix is square and is full rank.
An invertible matrix has only one vector in the null space: the zero vector. If any basis vector of a matrix is a linear transformation of another, then the matrix does not have basis and must be non-invertible.
For orthogonal matrices (such as permutation matrices), the inverse is also the transpose: Q-1 = QT.
Test with Determinant
The determinant is the most common test for invertibility. If |A| != 0, then A is invertible. If |A| = 0, then A is non-invertible.
Calculation with Determinant
Consider the 2 x 2 case:
┌ ┐ │ a b │ │ c d │ └ ┘
The inverse matrix is
1 ┌ ┐
―――――― │ d -b │
Det(A) │ -c a │
└ ┘
Calculation with Elimination
Because AA-1 = I, applying elimination and backwards elimination on A augmented with an identity matrix (I) will create A-1 in the augmentation.
┌ ┐ │ [1] 3 │ 1 0│ │ 2 7 │ 0 1│ └ ┘ ┌ ┐ │ [1] 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ ┌ ┐ │ 1 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ ┌ ┐ │ [1] 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘
A-1 is:
┌ ┐ │ 7 -3│ │ -2 1│ └ ┘
Calculation with Determinants and Cofactor Matrices
Given the determinant of A, it can also be simple to compute A-1 as (1/|A|)CT. C is the cofactor matrix, where ci j is the cofactor of ai j.
For example, given a 2 x 2 A like:
┌ ┐ │ a b│ │ c d│ └ ┘
The cofactor matrix C is:
┌ ┐ │ d -c│ │ -b a│ └ ┘
But this must be transposed to CT:
┌ ┐ │ d -b│ │ -c a│ └ ┘
And then A^-1 is:
┌ ┐ │ (1/det A) * d (1/det A) * -b│ │ (1/det A) * -c (1/det A) * a│ └ ┘
The above example fits into this formula. The elimination and backwards elimination prove that the determinant of that A is 1. The more fundamental formula ad - bc expands to 1 * 7 - 2 * 3 which also reveals a determinant of 1. As such, (1/|A|) is trivially 1. So simply plug the given (a, b, c, d) into the transposed cofactor matrix to find the inverse.