Tutorial

Rational Matrices

Most operations atlas uses involve integral matrices. However, rational matrices can be manipulated in atlas. In particular, they can be inverted directly without having to turn them into integral matrices. We need a special command because the command inverse only inverts matrices which are invertible over the integers. If they are not, we get an error

atlas> set A=mat:[[2,1],[0,1]]
Variable A: mat
atlas> A
Value:
| 2, 0 |
| 1, 1 |
atlas> inverse(A)
Runtime error:
  Matrix not invertible over the integers
(in call at atlas-scripts/basic.at:295:23-71 of error@string, built-in)
  [inv=
|  1, 0 |
| -1, 2 |
, d=2]
  [M=
| 2, 0 |
| 1, 1 |
]
(in call at <standard input>:8:0-10 of inverse@mat, defined at atlas-
scripts/basic.at:293:4--295:74)
Evaluation aborted.
atlas>

Instead we need to use the following:

atlas> set B=rational_inverse(A)
Variable B: (mat,string,int)
atlas> B
Value: (
|  1, 0 |
| -1, 2 |
,"/",2)
atlas>

As you can see a rational matrix is a triple that consists of an integral matrix, a string consisting of the division sign and an integer. What this means is that the integral matrix is divided by 2 to give the rational matrix which is the inverse of A

To print the matrix by itself we use the command show:

atlas> show(B)
1/2 -1/2
0 1
atlas>

We can multiply rational matrices with integral matrices and check in this case that we do have the inverse of A:

atlas> B*A
Value: (
| 1, 0 |
| 0, 1 |
,"/",1)
atlas>

But we can convert a rational matrix which is integral into its integral form:

atlas> set C=B*A
Variable C: (mat,string,int)
atlas>
Value: (
| 1, 0 |
| 0, 1 |
,"/",1)
atlas>
atlas> ratmat_as_mat (C)
Value:
| 1, 0 |
| 0, 1 |
atlas>

Rational matrices can also be multiplied with rational vectors and do other operations as with their integral counterparts:

atlas> B*[3/2, 1/2]
Value: [  3, -1 ]/4
atlas>