Skip to content Skip to navigation

Connexions

You are here: Home » Content » Matrix Inversion

Navigation

Lenses

What is a lens?

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? tag icon

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

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "State Space Systems"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Recently Viewed

This feature requires Javascript to be enabled.
 

Matrix Inversion

Module by: Thanos Antoulas, JP Slavinsky, Bailey Edwards. E-mail the authors

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=( a1,1…a1,n ⋮⋱⋮ an,1…an,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.

A⁢A-1=A-1⁢A=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=( α1,1…α1,n ⋮⋱⋮ αn,1…αn,n ) A α11 … α1n ⋮ ⋱ ⋮ αn1 … αnn
(3)
αi,j=-1i+j⁢det A ij αij -1 i j A ij
(4)

where A ij A ij is the (n−1)n1 x (n−1)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=1detA⁢AH 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=-12⁢d α11 -1 2 d
(7)
α1,2=-13⁢b α12 -1 3 b
(8)
α2,1=-13⁢c α21 -1 3 c
(9)
α2,2=-14⁢a α22 -1 4 a
(10)

Therefore,

AH=( d−b −ca ) A d b c a
(11)

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

A-1=1a⁢d−b⁢c⁢( d−b −ca ) A 1 a d b c d b c a
(12)

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

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? tag icon

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

| External bookmarks