Page 138 - Matrix Analysis & Applied Linear Algebra

132 Chapter 3 Matrix Algebra

These observations generalize to matrices of arbitrary size.
One of our objectives is to remove the arrows from Gaussian elimination
because the inability to do “arrow algebra” limits the theoretical analysis. For
example, while it makes sense to add two equations together, there is no mean-
ingful analog for arrows—reducing A → B and C → D by row operations does
not guarantee that A + C → B + D is possible. The following properties are
the mechanisms needed to remove the arrows from elimination processes.

Properties of Elementary Matrices
• When used as a left-hand multiplier, an elementary matrix of Type
I, II, or III executes the corresponding row operation.
• When used as a right-hand multiplier, an elementary matrix of Type
I, II, or III executes the corresponding column operation.

Proof. A proof for Type III operations is given—the other two cases are left to
the reader. Using I + αe j e T i as a left-hand multiplier on an arbitrary matrix A
produces

 
0 0 ··· 0
 . . . . . . 
 . .
T  .  th
I + αe j e A = A + αe j A i∗ = A + α  a i1 a i2 ··· a in  ← j row .

i


 . . . . . . 
. . .
0 0 ··· 0
This is exactly the matrix produced by a Type III row operation in which the
i th row of A is multiplied by α and added to the j th row. When I + αe j e T
i
is used as a right-hand multiplier on A, the result is
i th col
 
↓
0 ··· a 1j ··· 0
 
 0
T T ··· a 2j ··· 0 
A I + αe j e = A + αA ∗j e = A + α   .
i i  . .
 . . . . . 
.
. 
0 ··· a nj ··· 0
This is the result of a Type III column operation in which the j th column of A
is multiplied by α and then added to the i th column.