Page 234 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes How many elementary nilpotent blocks will there be in the rational form for N? The number
will be the nullity of N, i.e., the dimension of the characteristic space associated with the
characteristic value c. That dimension is 1, because any function which satisfies the differential
equation
Df = cf

cx
is a scalar multiple of the exponential function h(x) = e . Therefore, the operator N (on the
r–1
space V ) has a cyclic vector. A good choice for a cyclic vector is g = x h:
c
r–1 cx
g(x) = x e
This gives
r 
2
Ng Ng  Ng  (r  1)x  (r h 1)x r  2 (r  1)x r  2 h
h
     
h
!

N r  1 g N r  1 g N r  1 g  (r  1) (r  1)!h (r  1)!h
The preceding paragraph shows us that the Jordan form for D (on the space V) is the direct sum
of k elementary Jordan matrices, one for each root c .
i
Self Assessment
1. If A is an n × n matrix over the field F with characteristic polynomials
d
d
d
f = (x – c ) 1 (x – c ) 2 ... (x – c ) k
1 2 k
What is the trace of A?
2. Show that the matrix
 0 1 0 
 
A  1 0 1
 
  0  1 0  
is nilpotent. Show also that the Jordan form of A consists of a single 3 × 3 matrix.

21.3 Summary

 The findings of the theorem 1 of the last unit helps us to see that the finite vector space V
for a linear nilpotent operator is decomposed as the direct sum of its cyclic invariant
subspaces Z(; N) with N annihilators p , p , ..., p .
i 1 2 r
k
 Since N is nilpotent, the minimal polynomial is x where k  n, and thus each p is also of the
i
k
form p = x i .
i
 Theorem 1 of the last unit also helps us to write N as the direct sum of the elementary
nilpotent matrices known as companion matrices.
21.4 Keywords

Companion Matrix: is such an n × n matrix whose elements are zeros every where except
immediately below the diagonal line has 1s.
k
Nilpotent Matrix: A matrix A such that A = 0, is called nilpotent matrix of index k. Provided
k-1
a  0.

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