We introduce elementary matrices which have the following property: multiplying A
on the left by an elementary matrix has the effect of performing a row operation to A
.
Elementary matrices of type I
Definition:
Let i,j
be integers between 1
and n
inclusive, and assume i≠j
. Let λ∈𝐑
. Then Eij(λ)
is the matrix which has:
1s on the diagonal,
0s everywhere else except in the ij
th position, where there is a λ
.
Such a matrix is an elementary matrix of type I.
For example, if n=3
you get things like: E12(4)=(140010001),E32(-5)=(1000100-51).
If n=4
, you get things like E23(7)=(1000017000100001).
Lemma:
Let A
and Eij(λ)
be n
-by-n
matrices. Then Eij(λ)A
is the matrix obtained from A
by doing Ri↦Ri+λRj
.
Let's consider the case i<j
(the other case is similar so we omit it). Consider the product (1col icol j⋱↓↓row i→1λ⋱1⋱1)(A11⋯⋯⋯⋯A1n⋮⋮Ai1Ain⋮⋮Aj1Ajn⋮⋮An1⋯⋯⋯⋯Ann)
The only difference the λ
makes is when we multiply the i
th row into a column of A
(say the k
th column). Instead of just picking up 1×Aik
, we get 1×Aik+λ×Ajk
.
In other words, the result Eij(λ)A
is obtained from A
by adding λ
times row j
to row i
.
Elementary matrices of type II
Definition:
Let i
be an integer between 1
and n
inclusive and let λ≠0
be a real number. Define Ei(λ)
to be the matrix which has 1s on the diagonal everywhere except in the ii
th position, where it has a λ
. Such a matrix is an elementary matrix of type II.
If n=3
, E2(λ)=(1000λ0001),E3(-1)=(10001000-1).
Lemma:
Ei(λ)A
is obtained from A
by performing the row operation Ri↦λRi
.
In the product, (10⋯00⋱⋮λ⋮⋱00⋯01)(A11⋯A1n⋮⋮Ai1⋯Ain⋮⋮An1⋯Ann)
most of the entries are multiplied by 1
, except those in the i
th row of A
, which are rescaled by λ
.