simplify_full() for sage matrices

Question

My matrix is $$solution = \left(\begin{array}{rrr|r} 1 & 0 & 0 & \frac{2 \, {\left(\frac{2}{k - 2} - 3\right)}}{k - 1} - \frac{4}{k - 2} + 4 \\ 0 & 1 & 0 & \frac{2}{k - 2} - \frac{2}{{\left(k - 1\right)} {\left(k - 2\right)}} \\ 0 & 0 & 1 & \frac{2}{k - 1} \end{array}\right)$$

If I use the simplify command for an entry I get

sage: solution[0][3].simplify_full()

2*(2*k - 7)/(k - 1)

Is there a similar command for a full matrix?

Answer 1

Here's one way . . .

First define a function simp by

$\qquad$def simp(u): try: v=u.simplify_full() except: v=u return v

Then, for your matrix $A$, do

$\qquad$A = matrix(map(simp,A))

Answer 2

There is a built-in method for that, which is .simplify_full().

It can be applied to any matrix with entries in the symbolic ring.

To illustrate it, we define k as a symbolic variable, and build the matrix in the question.

sage: k = SR.var('k')
sage: a = identity_matrix(SR, 3)
sage: b = vector([
....:         (2 * (2 / (k - 2) - 3)) / (k - 1),
....:         2 / (k - 2) - 2 / ((k - 1) * (k - 2)),
....:         2 / (k - 1)
....:         ])
....: 
sage: m = a.augment(b, subdivide=True)

Then m is as follows:

sage: m
[                              1                               0                               0|      2*(2/(k - 2) - 3)/(k - 1)]
[                              0                               1                               0|2/(k - 2) - 2/((k - 1)*(k - 2))]
[                              0                               0                               1|                      2/(k - 1)]
sage: show(m)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r}
1 & 0 & 0 & \frac{2 \, {\left(\frac{2}{k - 2} - 3\right)}}{k - 1} \\
0 & 1 & 0 & \frac{2}{k - 2} - \frac{2}{{\left(k - 1\right)} {\left(k - 2\right)}} \\
0 & 0 & 1 & \frac{2}{k - 1}
\end{array}\right)

Applying the .simplify_full() method applies simplify_full to each entry, giving the expected result.

sage: mm = m.simplify_full()
sage: mm
[                           1                            0                            0 -2*(3*k - 8)/(k^2 - 3*k + 2)]
[                           0                            1                            0                    2/(k - 1)]
[                           0                            0                            1                    2/(k - 1)]

sage: show(mm)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
1 & 0 & 0 & -\frac{2 \, {\left(3 \, k - 8\right)}}{k^{2} - 3 \, k + 2} \\
0 & 1 & 0 & \frac{2}{k - 1} \\
0 & 0 & 1 & \frac{2}{k - 1}
\end{array}\right)

Providing this method for matrices with entries in the symbolic ring was the object of

• Sage trac ticket #12162: simplify_full for matrix

which was merged in the development release Sage 6.9.rc0 and in the stable release Sage 6.9.