matrix inverse

matrix inverse

1

Matrix Inverse

The

inverse

of an

n

×

n

matrix

A

is denoted by

A

-

1

. The inverse is defined so that

A

⁢

A

-

1

=

A

-

1

⁢

A

=

I

n

where

I

n

is the

n

×

n

identity matrix

.

It should be stressed that

only

square matrices

have inverses proper– however, a matrix of any size may have “left” and “right” inverses (which will not be discussed here).

A precondition for the

existence

of the

matrix inverse

A

-

1

(i.e. the matrix is

invertible

) is that

det

⁡

A

≠

0

(the

determinant

is nonzero), the reason for which we will see in a second.

The general form of the inverse of a matrix

A

is

A

-

1

=

1

det

⁡

(

A

)

⁢

adj

⁡

(

A

)

where

adj

⁡

(

A

)

is the

adjugate

of

A

(the matrix formed by the

cofactors

of

A

, i.e. with

adj

(

A

)

i

⁢

j

=

C

i

⁢

j

(

A

)

).

1

1

Some other sources call the adjugate the

adjoint

; however on PM the adjoint is reserved for the

conjugate transpose

.

This can also be thought of as a

generalization

of the

2

×

2

formula

given in the next

section

. However, due to the

inclusion

of the determinant in the expression, it is impractical to actually use this to calculate inverses.

This general form also explains why the determinant must be nonzero for invertibility; as we are dividing through by its value.

An

invertible matrix

is also said to be nonsingular.

2

Calculating the Inverse By Hand

Method 1:

An easy way to calculate the inverse of a matrix by hand is to form an augmented matrix

[

A

|

I

]

from

A

and

I

n

, then use

Gaussian elimination

to transform the left half into

I

. At the end of this procedure, the right half of the augmented matrix will be

A

-

1

(that is, you will be left with

[

I

|

A

-

1

]

).

Method 2:

One can calculate the

i

,

j

th element of the inverse by using the general formula; i.e.

A

j

⁢

i

-

1

=

C

i

⁢

j

⁢

(

A

)

/

det

⁡

A

where

C

i

⁢

j

⁢

(

A

)

is the

i

,

j

th cofactor expansion of the matrix

A

.

Note that the indices on the left-hand side are swapped relative to the right-hand side.

2.1

2-by-2 case:

For the

2

×

2

case, the general formula reduces to a memorable shortcut. For the

2

×

2

matrix

M

=

[

a

b

c

d

]

The inverse

M

-

1

is always

M

-

1

=

(

1

det

⁡

M

)

⁢

[

d

-

b

-

c

a

]

where

det

⁡

M

is simply

a

⁢

d

-

b

⁢

c

.

2.2

Remarks

Some caveats: computing the matrix inverse for

ill-conditioned matrices

is error-prone; special care must be taken and there are sometimes special algorithms to calculate the inverse of certain classes of matrices (for example,

Hilbert matrices

).

3

Avoiding the Inverse and Numerical Calculation

The need to find the matrix inverse depends on the situation– whether done by hand or by computer, and whether the matrix is simply a part of some equation or expression or not.

Instead of computing the matrix

A

-

1

as part of an equation or expression, it is nearly always better to use a

matrix factorization

instead. For example, when solving the system

A

⁢

x

=

b

, actually calculating

A

-

1

to get

x

=

A

-

1

⁢

b

is discouraged.

LU-factorization

is typically used instead.

We can even use this fact to speed up our calculation of the inverse by itself. We can cast the problem as finding

X

in

A

⁢

X

=

B

For

n

×

n

matrices

A

,

X

, and

B

(where

X

=

A

-

1

and

B

=

I

n

). To solve this, we first find the

L

⁢

U

decomposition

of

A

, then iterate over the columns, solving

L

⁢

y

=

P

⁢

b

k

and

U

⁢

x

k

=

y

each time (

k

=

1

⁢

…

⁢

n

). The resulting values for

x

k

are then the columns of

A

-

1

.

4

Elements of Invertible Matrices

Typically the

matrix elements

are members of a field when we are speaking of inverses (i.e. the reals, the

complex numbers

). However, the matrix inverse may exist in the case of the

elements

being members of a

commutative ring

, provided that the determinant of the matrix is a unit in the ring.

References

1

Golub and Van Loan, “Matrix Computations,” Johns Hopkins Univ. Press, 1996.

2

“Matrix Math,”

http://easyweb.easynet.co.uk/ mrmeanie/matrix/matrices.htm

http://easyweb.easynet.co.uk/ mrmeanie/matrix/matrices.htm

Title

matrix inverse

Canonical name

MatrixInverse

Date of creation

2013-03-22 13:04:25

Last modified on

2013-03-22 13:04:25

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

18

Author

akrowne (2)

Entry type

Definition

Classification

msc 15A09

Synonym

inverse

Synonym

matrix inversion

Defines

invertible

Defines

adjugate

Defines

adjugate of a matrix

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