Page 19 - Advanced Linear Algebra

Preliminaries 3

Block Matrices

It will be convenient to introduce the notational device of a block matrix. If ) Á
are matrices of the appropriate sizes, then by the block matrix

v ) ) Á Ä Á ) Á y
4~ Å Å Å
w ) ) Á Ä Á ) Á z block

we mean the matrix whose upper left submatrix is ) Á , and so on. Thus, the
) of 4 Á 's are submatrices and not entries. A square matrix of the form

v ) Ä y
x Æ Æ Å {
4~ x {
Å Æ Æ
w Ä ) z block

where each ) is square and is a zero submatrix, is said to be a block

diagonal matrix.
Elementary Row Operations
Recall that there are three types of elementary row operations. Type 1
operations consist of multiplying a row of ( by a nonzero scalar. Type 2
operations consist of interchanging two rows of . Type 3 operations consist of
(
adding a scalar multiple of one row of to another row of .
(
(
If we perform an elementary operation of type to an identity matrix 0 , the

result is called an elementary matrix of type . It is easy to see that all

elementary matrices are invertible.
we can perform
In order to perform an elementary row operation on ( C Á
0 , to obtain an elementary matrix and then take
,
that operation on the identity
the product ,( . Note that multiplying on the right by , has the effect of
performing column operations.
Definition A matrix is said to be in reduced row echelon form if
9
1 All rows consisting only of 's appear at the bottom of the matrix.
)

)
2 In any nonzero row, the first nonzero entry is a . This entry is called a

leading entry.
3 For any two consecutive rows, the leading entry of the lower row is to the
)
right of the leading entry of the upper row.
)

4 Any column that contains a leading entry has 's in all other positions.
Here are the basic facts concerning reduced row echelon form.