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Matrix Inversion

Summary: (Blank Abstract)

Let's say we have the square nn x nn matrix AA composed of real numbers. By "square", we mean it has the same number of rows as columns.

A=( a1,1a1,n an,1an,n ) A a11 a1n an1 ann
(1)

The subscripts of the real numbers in this matrix denote the row and column numbers, respectively (i.e. a1,2a12 holds the position at the intersection of the first row and the second column).

We will denote the inverse of this matrix as A-1 A . A matrix inverse has the property that when it is multiplied by the original matrix (on the left or on the right), the result will be the identity matrix.

AA-1=A-1A=I A A A A I
(2)

To compute the inverse of AA, two steps are required. Both involve taking determinants ( detA A ) of matrices. The first step is to find the adjoint ( AH A ) of the matrix A. It is computed as follows:

AH=( α1,1α1,n αn,1αn,n ) A α11 α1n αn1 αnn
(3)
αi,j=-1i+jdet A ij αij -1 i j A ij
(4)

where A ij A ij is the (n1)n1 x (n1)n1 matrix obtained from AA by eliminating its ii-th column and jj-th row. Note that we are not eliminating the ii-th row and jj-th column as you might expect.

To finish the process of determining the inverse, simply divide the adjoint by the determinant of the original matrix AA .

A-1=1detAAH A 1 A A
(5)

Example 1

A=( ab cd ) A a b c d
(6)

Find the inverse of the above matrix.

The first step is to compute the terms in the adjoint matrix:

α1,1=-12d α11 -1 2 d
(7)
α1,2=-13b α12 -1 3 b
(8)
α2,1=-13c α21 -1 3 c
(9)
α2,2=-14a α22 -1 4 a
(10)

Therefore,

AH=( db ca ) A d b c a
(11)

We then compute the determinant to be adbc a d b c . Dividing through by this quantity yields the inverse of AA:

A-1=1adbc( db ca ) A 1 a d b c d b c a
(12)

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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