Page 25 - Advanced Linear Algebra

Preliminaries 9

Example 0.3 Two matrices , ()  C ²-³ are row equivalent if and only if
there is an invertible matrix 7 such that ( ~ 7 ) . Similarly, and ) ( are
column equivalent, that is, can be reduced to using elementary column
(
)
operations, if and only if there exists an invertible matrix such that ( ~ ) 8 .
8
Two matrices ( and ) are said to be equivalent if there exist invertible
matrices and for which
7
8
(~ 7)8
Put another way, ( and ) are equivalent if ( can be reduced to ) by
performing a series of elementary row and/or column operations. The use of the
(
term equivalent is unfortunate, since it applies to all equivalence relations, not
just this one. However, the terminology is standard, so we use it here.)

It is not hard to see that an d matrix that is in both reduced row echelon
9
form and reduced column echelon form must have the block form
0 Á c
1~ > ?
c Á c Á c block
(
We leave it to the reader to show that every matrix in C is equivalent to
and so the set of these matrices is a set of
exactly one matrix of the form 1
canonical forms for equivalence. Moreover, the function defined by
²(³ ~ , where ( — 1 , is a complete invariant for equivalence.

1

Since the rank of is and since neither row nor column operations affect the
rank, we deduce that the rank of is . Hence, rank is a complete invariant for
(

equivalence. In other words, two matrices are equivalent if and only if they have
the same rank.
Example 0.4 Two matrices , ()  C ²-³ are said to be similar if there exists
an invertible matrix such that
7
(~ 7)7 c
. As we will learn,
Similarity is easily seen to be an equivalence relation on C
two matrices are similar if and only if they represent the same linear operators
on a given -dimensional vector space = . Hence, similarity is extremely

important for studying the structure of linear operators. One of the main goals of
this book is to develop canonical forms for similarity.
We leave it to the reader to show that the determinant function and the trace
function are invariants for similarity. However, these two invariants do not, in
general, form a complete system of invariants.

Example 0.5 Two matrices , ()  C ²-³ are said to be congruent if there
exists an invertible matrix for which
7