row reduction

row reduction

Row reduction

, also known as

Gaussian elimination

, is an algorithm for solving a

system of linear equations

a

11

⁢

x

1

+

a

12

⁢

x

2

+

…

+

a

1

⁢

m

⁢

x

m

=

b

1

a

21

⁢

x

1

+

a

22

⁢

x

2

+

…

+

a

2

⁢

m

⁢

x

m

=

b

2

⋮

⋮

⋱

⋮

⋮

a

n

⁢

1

⁢

x

1

+

a

n

⁢

2

⁢

x

2

+

…

+

a

n

⁢

m

⁢

x

m

=

b

n

To describe row reduction, it is convenient to formulate a

linear
system

as a single matrix-vector equation

A

⁢

x

=

b

,

where

A

=

[

a

11

a

12

⋯

a

1

⁢

m

a

21

a

22

⋯

a

2

⁢

m

⋮

⋮

⋱

⋮

a

n

⁢

1

a

n

⁢

2

⋯

a

n

⁢

m

]

,

b

=

[

b

1

b

2

⋮

b

n

]

,

x

=

[

x

1

x

2

⋮

x

m

]

are, respectively, the

n

×

m

matrix of

coefficients

of the
linear system, the

n

-place

column vector

of the scalars from the
right-hand of the equations, and the

m

-place column vector of

unknowns

.

The method consists of combining the coefficient matrix

A

with the
right hand vector

b

to form the “augmented”

n

×

(

m

+

1

)

matrix

[

A

b

]

=

[

a

11

a

12

⋯

a

1

⁢

m

b

1

a

21

a

22

⋯

a

2

⁢

m

b

2

⋮

⋮

⋱

⋮

⋮

a

n

⁢

1

a

n

⁢

2

⋯

a

n

⁢

m

b

n

]

=

[

R

1

R

2

⋮

R

n

,

]

where each

R

i

is the

m

+

1

-place

row vector

corresponding to row

i

of the augmented matrix.

A

sequence

of

elementary row operations

is then applied to this matrix
so as to transform it to

row echelon form

. The elementary operations are:

•

row scaling:

the

multiplication

a row by a nonzero
scalar;

R

i

↦

c

⁢

R

i

,

c

≠

0

;

•

row exchange:

the exchanges of two rows;

R

i

↔

R

j

;

•

row replacement:

the addition of a multiple of one row to
another row;

R

i

↦

R

i

+

c

⁢

R

j

,

c

≠

0

,

i

≠

j

.

Note that these

operations

are “legal” because

x

is a solution of
the transformed system if and only if it is a solution of the initial system.

If the number of equations equals the number of

variables

(

m

=

n

), and
if the coefficient matrix

A

is

non-singular

(

http://planetmath.org/

singular

), then the algorithm will
terminate when the augmented matrix has the following form:

[

a

11

′

a

12

′

⋯

a

1

⁢

n

′

b

1

0

a

22

′

⋯

a

2

⁢

n

′

b

2

′

⋮

⋮

⋱

⋮

⋮

0

0

⋯

a

n

⁢

n

′

b

n

′

]

With these

assumptions

, there exists a unique solution, which can be
obtained from the above matrix by back substitution.

For the general case, the termination procedure is somewhat more
complicated. First recall that a matrix is in

echelon form

if each
row has more leading zeros than the rows above it. A pivot is the
leading non-zero entry of some row. We then have

•

If there is a pivot in the last column, the system is

inconsistent

; there will be no solutions.

•

If that is not the case, then the general solution will have

d

degrees of freedom, where

d

is the number of columns from

1

to

m

that have no pivot. To be more precise, the general solution
will have the form of one particular solution plus an arbitrary

linear
combination

of

d

linearly independent

n

-vectors.

In even more prosaic

language

, the variables in the non-pivot
columns are to be considered “

free variables

” and should be
“moved” to the right-hand side of the equation. The general
solution is then obtained by arbitrarily choosing values of the free
variables, and then solving for the remaining “non-free” variables
that reside in the pivot columns.

A variant of Gaussian elimination is

Gauss-Jordan elimination

. In
this

variation

we reduce to echelon form, and then if the system
proves to be consistent, continue to apply the elementary row
operations until the augmented matrix is in

reduced echelon
form

. This means that not only does each pivot have all zeroes
below it, but that each pivot also has all zeroes above it.

In essence, Gauss-Jordan elimination performs the back substitution;
the values of the unknowns can be read off directly from the

terminal

augmented matrix. Not surprisingly, Gauss-Jordan elimination is
slower than Gaussian elimination. It is useful, however, for solving
systems on paper.

Title

row reduction

Canonical name

RowReduction

Date of creation

2013-03-22 12:06:48

Last modified on

2013-03-22 12:06:48

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

18

Author

rmilson (146)

Entry type

Algorithm

Classification

msc 15A06

Synonym

Gaussian elimination

Synonym

Gauss-Jordan elimination

Related topic

RowEchelonForm

Related topic

ReducedRowEchelonForm

Related topic

ElementaryMatrix

Defines

pivot

Defines

row operation

Defines

row exchange

Defines

row replacement

Defines

row scaling

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