|
Size: 2175
Comment: Fixed link
|
Size: 2682
Comment: Determinants
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 3: | Line 3: |
| Given a matrix ''A'' and an inverse matrix ''A^-1^'', the product is the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. This is an important property with several utilities. | 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'''. |
| Line 13: | Line 13: |
| 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]]. | An inverse matrix satisfies the equation '''''AA'''^-1^ = '''I'''''. |
| Line 15: | Line 15: |
| Not all matrices have an inverse matrix. If ''A^-1^'' exists, then A is '''invertible''' and '''non-singular'''. | [[LinearAlgebra/Determinants|Determinants]] are the test for '''invertability'''. if ''|'''A'''| != 0'', then '''''A''''' is invertable and non-singular. Conversely, if ''|'''A'''| = 0'', then '''''A''''' is singular and non-invertable. |
| Line 21: | Line 21: |
| For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrix]] ''P'', the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: ''P^-1^ = P^T^''. | 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. |
| Line 23: | Line 23: |
| For a square matrix ''A'', the '''left inverse''' is the same as the '''right inverse'''. ''AA^-1^ = A^-1^A = I'' | An invertible matrix has only one vector in the [[LinearAlgebra/NullSpaces|null space]]: the zero vector. For [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: '''''Q'''^-1^ = '''Q'''^T^''. For a square matrix '''''A''''', the '''left inverse''' is the same as the '''right inverse'''. '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I''''' |
| Line 31: | Line 35: |
| 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''). | 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'''''). |
| Line 43: | Line 47: |
| 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 calculated with [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reverse elimination]]. [[LinearAlgebra/Elimination#Simplification_with_Augmented_Matrices|Augment]] '''''A''''' with '''''I'''''. |
| Line 71: | Line 75: |
| ''A^-1^'' is: | '''''A'''^-1^'' is: |
Matrix Inversion
For some matrices A, the inverse matrix (A-1) is a matrix which can be multiplied by the original matrix to produce the identity matrix. The calculation of an inverse matrix, if it exists, is called inversion.
Contents
Definition
An inverse matrix satisfies the equation AA-1 = I.
Determinants are the test for invertability. if |A| != 0, then A is invertable and non-singular. Conversely, if |A| = 0, then A is singular and non-invertable.
Properties
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.
An invertible matrix has only one vector in the null space: the zero vector.
For orthogonal matrices (such as permutation matrices), the inverse is also the transpose: Q-1 = QT.
For a square matrix A, the left inverse is the same as the right inverse. AA-1 = A-1A = I
Calculation
Consider the below system, which shows an unknown matrix (A-1) multiplied by a known matrix (A) creating an identity matrix (I).
-1 A A = I ┌ ┐┌ ┐ ┌ ┐ │ 1 3││ a b│ │ 1 0│ │ 2 7││ c d│ = │ 0 1│ └ ┘└ ┘ └ ┘
The inverse matrix is calculated with elimination and reverse elimination. Augment A with I.
The elimination proceeds as:
┌ ┐ │ [1] 3 │ 1 0│ │ 2 7 │ 0 1│ └ ┘ ┌ ┐ │ [1] 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘
The reverse elimination proceeds as:
┌ ┐ │ 1 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ ┌ ┐ │ [1] 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘
A-1 is:
┌ ┐ │ 7 -3│ │ -2 1│ └ ┘