Page 67 - A Course in Linear Algebra with Applications

2.3: Elementary Matrices 51

Theorem 2.3.3
Let A be anmxn matrix. Then there exist elementary mxm
matrices E\,..., Ek and elementary nxn matrices F\,..., Fi
such that
E k---E lAF 1---F l=N,

the normal form of A.

Proof
By applying suitable row operations to A we can find elemen-
tary matrices Ei, ..., Ek such that B = Ek • • • E\A is in row
echelon form. Then column operations are applied to reduce
B to normal form; this procedure yields elementary matrices
t
Fi, ..., Fi such hat N = BF 1 • • • F = E k • • • E 1AF 1 • • • F t is
the normal form of A.
Corollary 2.3.4
For any matrix A there are invertible matrices X and Y such
1
1
that N = XAY, or equivalently A = X~ NY~ , where N is
the normal form of A.
For it is easy to see that every elementary matrix is in-
vertible; indeed the inverse matrix represents the inverse of the
corresponding elementary row (or column) operation. Since
by 1.2.3 any product of invertible matrices is invertible, the
corollary follows from 2.3.3.

Example 2.3.4

(1 2 2 \
Let A = „ „ , . Find the normal form N of A and write
\2 3 4 J
N as the product of A and elementary matrices as specified
in 2.3.3.
All we need do is to put A in normal form, while keeping
track of the elementary matrices that perform the necessary
row and column operations. Thus