Page 230 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
c  0  0 0
 1 c  0 0 
 
     ...(3)
 
 c 
 0 0  1 c 
 

each with c = c . Furthermore, the sizes of these matrices will decrease as one reads from left to
i
right. A matrix of the form (3) is called an elementary Jordan matrix with characteristic value c.
Now if we put all the bases for the W together, we obtain an ordered basis for V. Let us describe
i
the matrix A of T in this ordered basis.
The matrix A is the direct sum

 A 1 0  0 
 0 A  0 
A   2  ...(4)
    
  
 0 0 A k

of matrices A ,..., A . Each A is of the form
1 k i
J  ( ) i 0  0 
1
 ( ) i  
 0 J 2 0 
A 
i     
 
0 0  J ( ) l
  i n  
where each J ( ) i is an elementary Jordan matrix with characteristic value c . Also, within each A , i
j
i
the sizes of the matrices J ( ) i decrease as j increases. An n × n matrix A which satisfies all the
j
conditions described so far in this paragraph (for some distinct scalars c ,..., c ) will be said to be
1 k
in Jordan form.
We have just pointed out that if T is a linear operator for which the characteristic polynomial
factors completely over the scalar field, then there is an ordered basis for V in which T is
represented by a matrix which is in Jordan form. We should like to show now that this matrix is
something uniquely associated with T, up to the order in which the characteristic values of T are
written down.

The uniqueness we see as follows. Suppose there is some ordered basis for V in which T is
represented by the Jordan matrix A described in the previous paragraph. If A is a d × d matrix,
i i i
then d is clearly the multiplicity of c as a root of the characteristic polynomial for A, or for T. In
i i
other words, the characteristic polynomial for T is
d
d
f = (x – c ) 1 ... (x – c ) k
1 k
This shows that c , ..., c and d , ..., d are unique, up to the order in which we write them. The fact
j k 1 k
that A is the direct sum of the matrices A gives us a direct sum decomposition V = W  ...  W
i 1 k
invariant under T. Now note that W must be the null space of (T – c I) , where n = dim V; for,
n
i i
A – c I is clearly nilpotent and A – c I is non-singular for j  i. So we see that the subspaces W are
i i j i i
unique. If T is the operator induced on W by T, then the matrix A is uniquely determined as the
i i i
rational form for (T ... c I).
i i

224 LOVELY PROFESSIONAL UNIVERSITY