The elementary matrices result from doing a row operation to the identity matrix.
Definition 4.1.1The row operations consist of the following
Switch two rows.
Multiply a row by a nonzero number.
Replace a row by a multiple of another row added to it.
The elementary matrices are given in the following definition.
Definition 4.1.2The elementary matricesconsist of those matrices which result byapplying a row operation to an identity matrix. Those which involve switching rows of theidentity are called permutation matrices. More generally, if
(i1,i2,⋅⋅⋅,in)
is a permutation,a matrix which has a 1 in the i_{k}position in row k and zero in every other position of that rowis called a permutation matrix.Thus each permutation corresponds to a unique permutationmatrix.
As an example of why these elementary matrices are interesting, consider the following.
( ) ( ) ( )
0 1 0 a b c d x y z w
|( 1 0 0 |) |( x y z w |) = |( a b c d |)
0 0 1 f g h i f g h i
A 3 × 4 matrix was multiplied on the left by an elementary matrix which was obtained from row
operation 1 applied to the identity matrix. This resulted in applying the operation 1 to the given
matrix. This is what happens in general.
Now consider what these elementary matrices look like. First consider the one which involves
switching row i and row j where i < j. This matrix is of the form
The two exceptional rows are shown. The i^{th} row was the j^{th} and the j^{th} row was the i^{th} in the
identity matrix. Now consider what this does to a column vector.
Now denote by P^{ij} the elementary matrix which comes from the identity from switching rows i
and j. From what was just explained consider multiplication on the left by this elementary
matrix.
Lemma 4.1.3Let P^{ij}denote the elementary matrix which involves switching the i^{th}and the j^{th}rows. Then
ij
P A = B
where B is obtained from A by switching the i^{th}and the j^{th}rows.
As a consequence of the above lemma, if you have any permutation
(i,⋅⋅⋅,i )
1 n
, it follows from
Lemma 3.3.2 that the corresponding permutation matrix can be obtained by multiplying
finitely many permutation matrices, each of which switch only two rows. Now every
such permutation matrix in which only two rows are switched has determinant −1.
Therefore, the determinant of the permutation matrix for
(i1,⋅⋅⋅,in)
equals
(− 1)
^{p} where
the given permutation can be obtained by making p switches. Now p is not unique.
There are many ways to make switches and end up with a given permutation, but what
this shows is that the total number of switches is either always odd or always even.
That is, you could not obtain a given permutation by making 2m switches and 2k + 1
switches. A permutation is said to be even if p is even and odd if p is odd. This is an
interesting result in abstract algebra which is obtained very easily from a consideration of
elementary matrices and of course the theory of the determinant. Also, this shows that the
composition of permutations corresponds to the product of the corresponding permutation
matrices.
To see permutations considered more directly in the context of group theory, you should see a
good abstract algebra book such as [18] or [14].
Next consider the row operation which involves multiplying the i^{th} row by a nonzero constant,
c. The elementary matrix which results from applying this operation to the i^{th} row of the identity
matrix is of the form
this elementary matrix which multiplies the i^{th} row of the identity by
the nonzero constant, c. Then from what was just discussed and the way matrices are
multiplied,
( )
a11 a12 ⋅⋅⋅ ⋅⋅⋅ a1p
|| ... ... ... ||
|| ||
E (c,i)|| ai1 ai2 ⋅⋅⋅ ⋅⋅⋅ aip ||
|( ... ... ... |)
a a ⋅⋅⋅ ⋅⋅⋅ a
n1 n2 np
equals a matrix having the columns indicated below.
The case where i > j is handled similarly. This proves the following lemma.
Lemma 4.1.5Let E
(c× i+ j)
denote the elementary matrix obtained from I by replacing thej^{th}row with c times the i^{th}row added to it. Then
E(c× i+ j)A = B
where B is obtained from A by replacing the j^{th}row of A with itself added to c times the i^{th}row ofA.
The next theorem is the main result.
Theorem 4.1.6To perform any of the three row operations on a matrix A it suffices todo the row operation on the identity matrix obtaining an elementary matrix E and then takethe product, EA. Furthermore, each elementary matrix is invertible and its inverse is anelementary matrix.
Proof: The first part of this theorem has been proved in Lemmas 4.1.3 - 4.1.5. It only remains
to verify the claim about the inverses. Consider first the elementary matrices corresponding to row
operation of type three.
E(− c× i+ j)E (c ×i +j) = I
This follows because the first matrix takes c times row i in the identity and adds it to row j. When
multiplied on the left by E
(− c× i+ j)
it follows from the first part of this theorem that you take
the i^{th} row of E
(c× i+ j)
which coincides with the i^{th} row of I since that row was not changed,
multiply it by −c and add to the j^{th} row of E
(c ×i +j)
which was the j^{th} row of I
added to c times the i^{th} row of I. Thus E
(− c× i+ j)
multiplied on the left, undoes
the row operation which resulted in E
(c × i+ j)
. The same argument applied to the
product
E (c× i+ j)E (− c× i+ j)
replacing c with −c in the argument yields that this product is also equal to I. Therefore,
E
(c× i+ j)
^{−1} = E
(− c× i+ j)
.
Similar reasoning shows that for E
(c,i)
the elementary matrix which comes from multiplying
the i^{th} row by the nonzero constant, c,
( )
E (c,i)−1 = E c−1,i .
Finally, consider P^{ij} which involves switching the i^{th} and the j^{th} rows.
P ijPij = I
because by the first part of this theorem, multiplying on the left by P^{ij} switches the i^{th}
and j^{th} rows of P^{ij} which was obtained from switching the i^{th} and j^{th} rows of the
identity. First you switch them to get P^{ij} and then you multiply on the left by P^{ij}
which switches these rows again and restores the identity matrix. Thus