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. E-mail the author

Summary: How to invert matrices

Say that we have the following matrix and that we want to find its determinant.

A=( a1,1a1,2…a1,n a2,1a2,2…a2,n ⋮⋮⋱⋮ an,1an,2…an,n ) A a11 a12 … a1n a21 a22 … a2n ⋮ ⋮ ⋱ ⋮ an1 an2 … ann
(1)

Calculating the determinant of a matrix is a recursive process. Basically, we start by choosing any one row or column. The determinant will then be found with respect to this row or column. What this means is that we will find a sum of the products of this row or column's values and sub-determinants formed by blocking out the row and column of the particular value.

Why is this choice of row or column left to us instead of always being defined as, say, the first row? The reason is that by choosing this row or column wisely, we can sometimes reduce the amount of work we do. For example, if a certain row or column contains a few zeros, choosing it as the row/column that we take the determinant with respect to would be a smart move. As the values of this chosen row or column will be multiplied by sub-determinants of the matrix in question, a value of 00 in one of these products would mean that we have one less matrix whose determinant we need to calculate.

In the case of the matrix above, we'll compute the determinant with respect to the first column. The final equation for the determinant is:

detA=a1,1⁢-11+1⁢detA11+a2,1⁢-12+1⁢detA12+…+an,1⁢-1n+1⁢detA1n A a11 -1 1 1 A11 a21 -1 2 1 A12 … an1 -1 n 1 A1n
(2)

Here, AijAij means the matrix formed by eliminating the ii-th column and the jj-th row of AA.

Let's just look at the first term in final equation for determinant. It is basically the first element of AA's first column times the determinant of the matrix formed by the elimination of the first row and first column of AA. There is also a −1r+c1rc term included. This serves to make the signs of all of the terms in the determinant equation fluctuate back and forth. The next term is the same, except that we have moved on to the second element in the first column of AA. As this element holds a position in the second row and first column of AA, the sub-determinant in this term is obtained by hiding the second row and first column of AA.

In a generic 33 x 33 example, we would find the following solution for the determinant:

det( a1,1a1,2a1,3 a2,1a2,2a2,3 a3,1a3,2a3,3 )=a1,1⁢det( a2,2a2,3 a3,2a3,3 )−a2,1⁢det( a1,2a1,3 a3,2a3,3 )+a3,1⁢det( a1,2a1,3 a2,2a2,3 ) a11 a12 a13 a21 a22 a23 a31 a32 a33 a11 a22 a23 a32 a33 a21 a12 a13 a32 a33 a31 a12 a13 a22 a23
(3)

To find the determinants of the 22 x 22 sub-determinants, we could again apply the rule of the final equation for determinant, keeping in mind that the determinant of a scalar value is simply that scalar value. However, it is easier to remember the following solution

det( ab cd )=a⁢d−b⁢c a b c d ad bc
(4)

Example 1

To clarify, take the following example of finding the determinant of a numeric 33 x 33 matrix.

A=( 1-12 311 -2-20 ) A 1 -1 2 3 1 1 -2 -2 0
(5)

First we need to choose a row or column to take the determinant with respect to. We notice that the element in the third row and third column is a zero. Knowing that choosing a row or column that contains a zero will reduce our workload, we will choose the third column. Then, by applying final equation for determinant, we get

detA=2×-14⁢det( 31 -2-2 )+1×-15⁢det( 1-1 -2-2 )+0 A 2 -1 4 3 1 -2 -2 1 -1 5 1 -1 -2 -2 0
(6)

Content actions

Download module as:

PDF | 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