Page 237 - DMTH502_LINEAR_ALGEBRA

Unit 22: Computation of Invariant Factors

In this case Notes
x 0 0  0 C 0
1 x 0  0 C 1
0 1 x  0 C 2
xI – A =
     
0 0 0   C n 2
0 0 0  1 x C n 1
In the row-operation, let us add x times row n to row (n – 1). This will remove the x in the (n – 1,
n – 1) place and still the determinant of [xI – A] does not change. To continue, add x times the new
row (n – 1) to row (n – 2). Continuing successively unit all of the x’s on the main diagonal have
been removed by that process, the result is the matrix

0 0 0  0 x n ... C x C 0
1
1 0 0  0 x n 1 ... C x C 1
2
0 1 0  0 x n 2 ... C x C 2
3
          
0 0 0  0 x 2 C n 1 x C n 2
0 0 0  01 x C n 1

which has the same determinant as xI – A. The upper right-hand entry of this matrix is the
polynomial p. Now we use column operations to clean up the last columns. We do so by adding
to last column appropriate multiples of the other columns:

0 0 0  0 p
1 0 0  0 0
0 1 0  0 0
   
0 0 0  0 0
0 0 0  1 0

Multiply each of the first (n – 1) columns by –1 and then perform (n – 1) interchanges of adjacent
columns to bring the present nth column to the first position. The total effect of the 2n – 2 sign
changes is to have the determinant unaltered. We obtain the matrix

1 0 0  0
0 1 0  0
0 0 1  0
    ...(1)
0 0 0  1

It is then clear that p = det (xI – A).

22.2 Computation of Invariant Factors

We are going to show that for any n n matrix A, there is a succession of row and column
operations which will transform xI – A into a matrix, in which the invariant factors of A appear
down the main diagonal.

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