matrix-finite-ring.sagews

Matrix ring over finite ring

http://doc.sagemath.org/html/en/constructions/rings.html#matrix-rings

R = IntegerModRing(10)
R

Ring of integers modulo 10

M = MatrixSpace(R, 4, 4)
M

Full MatrixSpace of 4 by 4 dense matrices over Ring of integers modulo 10

notice, that 99 ends up being 9

m = M([[1, 2, 3, 4],
       [4, 1,99, 1],
       [0, 8, 1, 8],
       [0, 0, 1, 2]])
m

[1 2 3 4] [4 1 9 1] [0 8 1 8] [0 0 1 2]

m - 1

[0 2 3 4] [4 0 9 1] [0 8 0 8] [0 0 1 1]

m + m

[2 4 6 8] [8 2 8 2] [0 6 2 6] [0 0 2 4]

m * m

[9 8 8 8] [8 1 1 1] [2 6 1 2] [0 8 3 2]

m^3

[1 0 5 4] [2 5 5 3] [6 8 3 6] [2 2 7 6]

M(1)

[1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1]

2 * m  + 9 * M(1)

[1 4 6 8] [8 1 8 2] [0 6 1 6] [0 0 2 3]

m2 = m.apply_map(lambda x : 2*x + 1)
m2

[3 5 7 9] [9 3 9 3] [1 7 3 7] [1 1 3 5]

m3 = 2 * m + M(1) - m2
m3

[0 9 9 9] [9 0 9 9] [9 9 0 9] [9 9 9 0]

m3.is_symmetric()

True