Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
The right eigenvectors are nothing but the left eigenvectors of the transpose matrix:
sage: left = A.transpose().eigenvectors_left(); left
[(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17
sage: right[0][1] == left[0][1]
True
exp()
Return the matrix exponential of this matrix X, which is the matrix
e X =
∞∑
k=0
This function depends on maxima’s matrix exponentiation function, which does not deal well with floating
point numbers. If the matrix has floating point numbers, they will be rounded automatically to rational
numbers during the computation.
EXAMPLES:
X k
k! .
sage: m = matrix(SR,2, [0,x,x,0]); m
[0 x]
[x 0]
sage: m.exp()
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]
sage: exp(m)
[1/2*(e^(2*x) + 1)*e^(-x) 1/2*(e^(2*x) - 1)*e^(-x)]
[1/2*(e^(2*x) - 1)*e^(-x) 1/2*(e^(2*x) + 1)*e^(-x)]
Exp works on 0x0 and 1x1 matrices:
sage: m = matrix(SR,0,[]); m
[]
sage: m.exp()
[]
sage: m = matrix(SR,1,[2]); m
[2]
sage: m.exp()
[e^2]
Commuting matrices m, n have the property that e m+n = e m e n (but non-commuting matrices need not):
sage: m = matrix(SR,2,[1..4]); n = m^2
sage: m*n
[ 37 54]
[ 81 118]
sage: n*m
[ 37 54]
[ 81 118]
sage: a = exp(m+n) - exp(m)*exp(n)
sage: a.simplify_rational() == 0
True
The input matrix must be square:
sage: m = matrix(SR,2,3,[1..6]); exp(m)
Traceback (most recent call last):
...
ValueError: exp only defined on square matrices
308 Chapter 16. Symbolic matrices