1 | initial version |

Thank you for pointing out this problem.

The computation of symbolic powers of a matrix was recently introduced in Sage, based on an idea proposed on Ask Sage:

- compute the Jordan form, and the transformation matrix
- compute powers of each Jordan block
- form a block-diagonal matrix with these powers
- change back to the old basis using the transformation matrix

The idea is good. Unfortunately there was a glitch in the implementation: the powers of the Jordan blocks were being inserted at the wrong place.

To illustrate the bug in an enlightening way, consider the matrix:

```
sage: n = SR.var('n')
sage: A = matrix(QQ, 3, [[2, 1, 0], [0, 2, 0], [0, 0, 3]])
sage: A
[2 1 0]
[0 2 0]
[0 0 3]
```

The current implementation in Sage gives:

```
sage: B = A^n; B
[ 2^n 2^(n - 1)*n 0]
[ 0 3^n 0]
[ 0 0 0]
```

Block k is inserted starting at position (k, k) instead of its correct position.

The revised implementation, suggested below, gives:

```
sage: B = _matrix_power_symbolic(A, n); B
[ 2^n 2^(n - 1)*n 0]
[ 0 2^n 0]
[ 0 0 3^n]
```

and can deal with the matrix in the question here.

```
sage: A = matrix([[4, 1, 2], [0, 2, -4], [0, 1, 6]])
sage: A
[ 4 1 2]
[ 0 2 -4]
[ 0 1 6]
sage: B = _matrix_power_symbolic(A, n); B
[ 4^n 4^(n - 1)*n 2*4^(n - 1)*n]
[ 0 -2*4^(n - 1)*n + 4^n -4*4^(n - 1)*n]
[ 0 4^(n - 1)*n 2*4^(n - 1)*n + 4^n]
```

Here is a revised implementation with a fix for the glitch, along with some efficiency improvements.

```
def _matrix_power_symbolic(A, n):
r"""
Return the symbolic matrix power `A^n`
This function implements the computation of `A^n` for symbolic `n`,
relying on the Jordan normal form of `A`, available for exact rings
as :meth:`jordan_form`. See [Hig2008]_, §1.2, for further details.
INPUT:
- ``A`` -- a square matrix over an exact field
- ``n`` -- the symbolic exponent
OUTPUT:
The matrix `A^n` (with symbolic entries).
EXAMPLES:
Powers of two by two matrix::
sage: n = SR.var('n')
sage: A = matrix(QQ, [[2, -1], [1, 0]])
sage: B = A^n
sage: B
[ n + 1 -n]
[ n -n + 1]
sage: all(A^k == B.subs({n: k}) for k in range(8))
True
Powers of a three by three matrix in Jordan form::
sage: n = SR.var('n')
sage: A = matrix(QQ, 3, [[2, 1, 0], [0, 2, 0], [0, 0, 3]])
sage: A
[2 1 0]
[0 2 0]
[0 0 3]
sage: B = A^n; B
[ 2^n 2^(n - 1)*n 0]
[ 0 2^n 0]
[ 0 0 3^n]
sage: all(A^k == B.subs({n: k}) for k in range(8))
True
Powers of a three by three matrix not in Jordan form::
sage: A = matrix([[4, 1, 2], [0, 2, -4], [0, 1, 6]])
sage: A
[ 4 1 2]
[ 0 2 -4]
[ 0 1 6]
sage: B = A^n
sage: B
[ 4^n 4^(n - 1)*n 2*4^(n - 1)*n]
[ 0 -2*4^(n - 1)*n + 4^n -4*4^(n - 1)*n]
[ 0 4^(n - 1)*n 2*4^(n - 1)*n + 4^n]
sage: [B.subs({n: k}) for k in range(4)]
[
[1 0 0] [ 4 1 2] [ 16 8 16] [ 64 48 96]
[0 1 0] [ 0 2 -4] [ 0 0 -32] [ 0 -32 -192]
[0 0 1], [ 0 1 6], [ 0 8 32], [ 0 48 160]
]
sage: all(A^k == B.subs({n: k}) for k in range(8))
True
TESTS:
Testing exponentiation in the symbolic ring::
sage: n = var('n')
sage: A = matrix([[pi, e],[0, -2*I]])
sage: (A^n).list()
[pi^n,
-(-2*I)^n/(pi*e^(-1) + 2*I*e^(-1)) + pi^n/(pi*e^(-1) + 2*I*e^(-1)),
0,
(-2*I)^n]
If the base ring is inexact, the Jordan normal form is not available::
sage: A = matrix(RDF, [[2, -1], [1, 0]])
sage: A^n
Traceback (most recent call last):
...
ValueError: Jordan normal form not implemented over inexact rings.
Testing exponentiation in the integer ring::
sage: A = matrix(ZZ, [[1,-1],[-1,1]])
sage: A^(2*n+1)
[ 1/2*2^(2*n + 1) -1/2*2^(2*n + 1)]
[-1/2*2^(2*n + 1) 1/2*2^(2*n + 1)]
Check if :trac:`23215` is fixed::
sage: a, b, k = var('a, b, k')
sage: (matrix(2, [a, b, -b, a])^k).list()
[1/2*(a + I*b)^k + 1/2*(a - I*b)^k,
-1/2*I*(a + I*b)^k + 1/2*I*(a - I*b)^k,
1/2*I*(a + I*b)^k - 1/2*I*(a - I*b)^k,
1/2*(a + I*b)^k + 1/2*(a - I*b)^k]
"""
from sage.rings.qqbar import AlgebraicNumber
from sage.matrix.constructor import matrix
from sage.functions.other import binomial
from sage.symbolic.ring import SR
from sage.rings.qqbar import QQbar
got_SR = A.base_ring() == SR
# Change to QQbar if possible
try:
A = A.change_ring(QQbar)
except (TypeError, NotImplementedError):
pass
# Get Jordan matrix J and invertible matrix P such that A = P*J*~P
# From that, we will compute M = J^n, and obtain A^n = P*J^n*~P
J, P = A.jordan_form(transformation=True)
# Where each Jordan block starts, and number of blocks
block_start = [0] + J.subdivisions()[0]
num_blocks = len(block_start)
# Prepare matrix M to store `J^n`, computed by Jordan block
M = matrix(SR, J.ncols())
M.subdivide(J.subdivisions())
for k in range(num_blocks):
# Jordan block Jk, its dimension nk, the eigenvalue m
Jk = J.subdivision(k, k)
nk = Jk.ncols()
mk = Jk[0,0]
# First row of block Mk; its entries are of the form
# D^i(f) / i! with f = x^n and D = differentiation wrt x
if hasattr(mk, 'radical_expression'):
mk = mk.radical_expression()
vk = [(binomial(n, i) * mk**(n-i)).simplify_full()
for i in range(nk)]
# Form block Mk and insert it in M
Mk = matrix(SR, [[SR.zero()]*i + vk[:-i] for i in range(nk)])
M.set_block(block_start[k], block_start[k], Mk)
# Change entries of P and P^-1 into symbolic expressions
if not got_SR:
Pinv = (~P).apply_map(AlgebraicNumber.radical_expression)
P = P.apply_map(AlgebraicNumber.radical_expression)
else:
Pinv = ~P
return P * M * Pinv
```

I will open a ticket on Sage's issue tracker to get the fix in Sage.

2 | No.2 Revision |

Thank you for pointing out this problem.

The computation of symbolic powers of a matrix was recently
introduced in ~~Sage, ~~Sage, based on an ~~idea ~~idea
proposed on Ask ~~Sage:~~Sage:

- compute the Jordan form, and the transformation matrix
- compute powers of each Jordan block
- form a block-diagonal matrix with these powers
- change back to the old basis using the transformation matrix

The idea is good. Unfortunately there was a glitch in the implementation: the powers of the Jordan blocks were being inserted at the wrong place.

To illustrate the bug in an enlightening way, consider the matrix:

```
sage: n = SR.var('n')
sage: A = matrix(QQ, 3, [[2, 1, 0], [0, 2, 0], [0, 0, 3]])
sage: A
[2 1 0]
[0 2 0]
[0 0 3]
```

The current implementation in Sage gives:

```
sage: B = A^n; B
[ 2^n 2^(n - 1)*n 0]
[ 0 3^n 0]
[ 0 0 0]
```

Block k is inserted starting at position (k, k) instead of its correct position.

The revised implementation, suggested below, gives:

```
sage: B = _matrix_power_symbolic(A, n); B
[ 2^n 2^(n - 1)*n 0]
[ 0 2^n 0]
[ 0 0 3^n]
```

and can deal with the matrix in the question here.

```
sage: A = matrix([[4, 1, 2], [0, 2, -4], [0, 1, 6]])
sage: A
[ 4 1 2]
[ 0 2 -4]
[ 0 1 6]
sage: B = _matrix_power_symbolic(A, n); B
[ 4^n 4^(n - 1)*n 2*4^(n - 1)*n]
[ 0 -2*4^(n - 1)*n + 4^n -4*4^(n - 1)*n]
[ 0 4^(n - 1)*n 2*4^(n - 1)*n + 4^n]
```

Here is a revised implementation with a fix for the glitch, along with some efficiency improvements.

```
def _matrix_power_symbolic(A, n):
r"""
Return the symbolic matrix power `A^n`
This function implements the computation of `A^n` for symbolic `n`,
relying on the Jordan normal form of `A`, available for exact rings
as :meth:`jordan_form`. See [Hig2008]_, §1.2, for further details.
INPUT:
- ``A`` -- a square matrix over an exact field
- ``n`` -- the symbolic exponent
OUTPUT:
The matrix `A^n` (with symbolic entries).
EXAMPLES:
Powers of two by two matrix::
sage: n = SR.var('n')
sage: A = matrix(QQ, [[2, -1], [1, 0]])
sage: B = A^n
sage: B
[ n + 1 -n]
[ n -n + 1]
sage: all(A^k == B.subs({n: k}) for k in range(8))
True
Powers of a three by three matrix in Jordan form::
sage: n = SR.var('n')
sage: A = matrix(QQ, 3, [[2, 1, 0], [0, 2, 0], [0, 0, 3]])
sage: A
[2 1 0]
[0 2 0]
[0 0 3]
sage: B = A^n; B
[ 2^n 2^(n - 1)*n 0]
[ 0 2^n 0]
[ 0 0 3^n]
sage: all(A^k == B.subs({n: k}) for k in range(8))
True
Powers of a three by three matrix not in Jordan form::
sage: A = matrix([[4, 1, 2], [0, 2, -4], [0, 1, 6]])
sage: A
[ 4 1 2]
[ 0 2 -4]
[ 0 1 6]
sage: B = A^n
sage: B
[ 4^n 4^(n - 1)*n 2*4^(n - 1)*n]
[ 0 -2*4^(n - 1)*n + 4^n -4*4^(n - 1)*n]
[ 0 4^(n - 1)*n 2*4^(n - 1)*n + 4^n]
sage: [B.subs({n: k}) for k in range(4)]
[
[1 0 0] [ 4 1 2] [ 16 8 16] [ 64 48 96]
[0 1 0] [ 0 2 -4] [ 0 0 -32] [ 0 -32 -192]
[0 0 1], [ 0 1 6], [ 0 8 32], [ 0 48 160]
]
sage: all(A^k == B.subs({n: k}) for k in range(8))
True
TESTS:
Testing exponentiation in the symbolic ring::
sage: n = var('n')
sage: A = matrix([[pi, e],[0, -2*I]])
sage: (A^n).list()
[pi^n,
-(-2*I)^n/(pi*e^(-1) + 2*I*e^(-1)) + pi^n/(pi*e^(-1) + 2*I*e^(-1)),
0,
(-2*I)^n]
If the base ring is inexact, the Jordan normal form is not available::
sage: A = matrix(RDF, [[2, -1], [1, 0]])
sage: A^n
Traceback (most recent call last):
...
ValueError: Jordan normal form not implemented over inexact rings.
Testing exponentiation in the integer ring::
sage: A = matrix(ZZ, [[1,-1],[-1,1]])
sage: A^(2*n+1)
[ 1/2*2^(2*n + 1) -1/2*2^(2*n + 1)]
[-1/2*2^(2*n + 1) 1/2*2^(2*n + 1)]
Check if :trac:`23215` is fixed::
sage: a, b, k = var('a, b, k')
sage: (matrix(2, [a, b, -b, a])^k).list()
[1/2*(a + I*b)^k + 1/2*(a - I*b)^k,
-1/2*I*(a + I*b)^k + 1/2*I*(a - I*b)^k,
1/2*I*(a + I*b)^k - 1/2*I*(a - I*b)^k,
1/2*(a + I*b)^k + 1/2*(a - I*b)^k]
"""
from sage.rings.qqbar import AlgebraicNumber
from sage.matrix.constructor import matrix
from sage.functions.other import binomial
from sage.symbolic.ring import SR
from sage.rings.qqbar import QQbar
got_SR = A.base_ring() == SR
# Change to QQbar if possible
try:
A = A.change_ring(QQbar)
except (TypeError, NotImplementedError):
pass
# Get Jordan matrix J and invertible matrix P such that A = P*J*~P
# From that, we will compute M = J^n, and obtain A^n = P*J^n*~P
J, P = A.jordan_form(transformation=True)
# Where each Jordan block starts, and number of blocks
block_start = [0] + J.subdivisions()[0]
num_blocks = len(block_start)
# Prepare matrix M to store `J^n`, computed by Jordan block
M = matrix(SR, J.ncols())
M.subdivide(J.subdivisions())
for k in range(num_blocks):
# Jordan block Jk, its dimension nk, the eigenvalue m
Jk = J.subdivision(k, k)
nk = Jk.ncols()
mk = Jk[0,0]
# First row of block Mk; its entries are of the form
# D^i(f) / i! with f = x^n and D = differentiation wrt x
if hasattr(mk, 'radical_expression'):
mk = mk.radical_expression()
vk = [(binomial(n, i) * mk**(n-i)).simplify_full()
for i in range(nk)]
# Form block Mk and insert it in M
Mk = matrix(SR, [[SR.zero()]*i +
```~~vk[:-i] ~~vk[:nk-i] for i in range(nk)])
M.set_block(block_start[k], block_start[k], Mk)
# Change entries of P and P^-1 into symbolic expressions
if not got_SR:
Pinv = (~P).apply_map(AlgebraicNumber.radical_expression)
P = P.apply_map(AlgebraicNumber.radical_expression)
else:
Pinv = ~P
return P * M * Pinv

I will open a ticket on Sage's issue tracker to get the fix in Sage.

Edit: The issue is now tracked at Sage Trac ticket #25082.

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