Page 245 - DMTH502_LINEAR_ALGEBRA

Unit 22: Computation of Invariant Factors

22.3 Summary Notes

 In this unit a method for computing the invariant factors p ... p which define the rational
1 r
form of the matrix, is given. It is shown that by elementary row and column operations it
can be achieved.
 It is shown that if N is row-equivalent to a matrix M then N = PM where p is a product of
elementary matrices.
 By this method one can show that
xI A 1 0  0
0 xI A 2  0
–1
–1
P(xI – A) P = xI – PAP = 
0   xI A r
where A is companion matrix.
i
22.4 Keywords

n n
An Elementary Matrix in F ( ) is one which can be obtained from n n identity matrix by
x
means of a single elementary operation.
An Elementary Row Operation: An elementary row operation on a matrix M whose determinant
has to be found, will not change the determinant of M if this row operation is one of the
following: (i) multiplication of one row of M by a non-zero scalar in F; (ii) replacement of the rth
row of M by the row r plus f times row s, where f is any polynomial over F and r s; (iii)
interchange of two rows of M.
m m
Row equivalent: Let M, N be matrices in F ( x . We say that N is row equivalent to M if N can
)
be obtained from M by a finite succession of elementary row operations.

22.5 Review Question

8
1. Let T be the linear operator on R which is represented in the standard basis by the matrix
1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1
0 1 1 0 0 0 0 1
A =
0 0 0 1 1 0 0 0
0 1 1 1 1 1 0 1
0 1 1 1 1 0 1 1
0 0 0 0 0 0 0 0
(a) Find the characteristic polynomial and the invariant factors.
(b) Find the Jordan form of A.

8
(c) Find a direct sum decomposition of R into T-cyclic subspaces as in theorem 1 of
unit 20.

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