|
Size: 1643
Comment:
|
Size: 2431
Comment: Reorganized
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 2: | Line 2: |
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 9: | Line 11: |
| == Introduction == | == Definition == |
| Line 11: | Line 13: |
| An '''inverse matrix''' is a matrix A^-1^ where multiplying it by matrix A results in the identity matrix. | An inverse matrix satisfies the equation '''''AA'''^-1^ = '''I'''''. |
| Line 13: | Line 15: |
| If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. ---- |
Not all matrices have an inverse that can satisfy that condition. If '''''A'''^-1^'' exists, then '''''A''''' is '''invertible''' and '''non-singular'''. |
| Line 19: | Line 19: |
| == Notable Properties == | === Properties === |
| Line 21: | Line 21: |
| For a permutation matrix, the inverse is also the 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 | For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrix]] '''''P''''', the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: '''''P'''^-1^ = '''P'''^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 33: |
| Consider the below problem: | 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 34: | Line 36: |
| -1 A A = I |
|
| Line 40: | Line 45: |
| 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 41: | Line 47: |
=== Gauss-Jordan Calculation === The inverse matrix can be calculated through elimination and reverse elimination. First step: |
The elimination proceeds as: |
| Line 53: | Line 54: |
2 - 1m = 0 m = 2 2 7 0 1 - 1m - 3m - 1m - 0m ____ ____ ____ ____ 0 1 -2 1 ┌ ┐ │ [1] 3 │ 1 0│ │ 0 1 │ -2 1│ └ ┘ |
┌ ┐ │ [1] 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ |
| Line 68: | Line 60: |
| Second step: | The reverse elimination proceeds as: |
| Line 75: | Line 67: |
3 - 1m = 0 m = 3 1 3 1 0 - 0m - 1m - -2m - 1m ____ ____ _____ ____ 1 0 7 -3 ┌ ┐ │ 1 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘ |
┌ ┐ │ [1] 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘ |
| Line 90: | Line 73: |
| The inverse matrix of A 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.
Not all matrices have an inverse that can satisfy that condition. If A-1 exists, then A is invertible and non-singular.
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.
For a permutation matrix P, the inverse is also the transpose: P-1 = PT.
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│ └ ┘