- Is it possible to predict how much time it will take to give a result by
Det[M]? - Is it possible to decompose the matrix (as a multiplication of simpler matrices) so that computing its determinant is faster? Can this be done using automated functionality provided by Mathematica?
Is it possible to decompose the given matrix as a multiplication Automated decomposition of simpler matrices so that computing their determinant isa large symbolic matrix for faster? determinant calculation
Is it possible to decompose the given matrix as a multiplication of simpler matrices so that computing their determinant is faster?
- Is it possible to predict how much time it will take to give a result by
Det[M]? - Is it possible to decompose the matrix (as a multiplication of simpler matrices) so that computing its determinant is faster?
Automated decomposition of a large symbolic matrix for faster determinant calculation
- Is it possible to predict how much time it will take to give a result by
Det[M]? - Is it possible to decompose the matrix (as a multiplication of simpler matrices) so that computing its determinant is faster? Can this be done using automated functionality provided by Mathematica?
Is it possible to decompose the given matrix as a multiplication of simpler matrices so that computing their determinant is faster?
I have this $44$ dimensional parametric matrix ($\{x,d,b,t\}\in\mathbb{R}$) and I want to compute its determinant.
Questions:
- Is it possible to predict how much time it will take to give a result by
Det[M]? - Is it possible to decompose the matrix (as a multiplication of simpler matrices) so that computing its determinant is faster?
M:={{-E^(-((I x)/2)) (1 + x), E^((I x)/2) (-1 + x), 0,
0, -E^(I b - (2 I d \[Pi])/11 - I x) (-1 + x),
E^(-(2/11) I d \[Pi] + I (b + x)) (1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0}, {0, 0, 1 + x,
1 - x, -E^(I b - (2 I d \[Pi])/11 - I x) (1 + x),
E^(-(2/11) I d \[Pi] + I (b + x)) (-1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0}, {0, 0, -1 + x, -1 - x, 1 + x, 1 - x, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-E^(-((I x)/2)) (-1 + x),
E^((I x)/2) (1 + x), 0, 0, -1 + x, -1 - x, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0,
0}, {E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0, 0, 0,
E^((4 I d \[Pi])/11 + I (-b + x)) (1 + x),
E^((4 I d \[Pi])/11 - I (b + x)) (1 - x), 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0,
E^((4 I d \[Pi])/11 + I (-b + x)) (-1 + x),
E^((4 I d \[Pi])/11 - I (b + x)) (-1 - x), 0,
0, -E^(-((I x)/2)) (-1 + x), E^((I x)/2) (1 + x), 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0}, {0, 0, 0, 0, 0, 0, 1 - x, 1 + x, 0,
0, -E^(-((I x)/2)) (1 + x), E^((I x)/2) (-1 + x), 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0}, {E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0, 0, 0,
0, -1 - x, -1 + x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0,
0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (1 + x), E^((I x)/2) (-1 + x), 0,
0, -E^(I b - (6 I d \[Pi])/11 - I x) (-1 + x),
E^(-(6/11) I d \[Pi] + I (b + x)) (1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), -E^(
I b - (6 I d \[Pi])/11 - I x) (1 + x),
E^(-(6/11) I d \[Pi] + I (b + x)) (-1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 1 + x, 1 - x, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (-1 + x),
E^((I x)/2) (1 + x), 0, 0, -1 + x, -1 - x, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0, 0, 0, 0, 0,
E^((8 I d \[Pi])/11 + I (-b + x)) (1 + x),
E^((8 I d \[Pi])/11 - I (b + x)) (1 - x), 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (-1 + x),
E^((I x)/2) (1 + x), E^((8 I d \[Pi])/11 + I (-b + x)) (-1 + x),
E^((8 I d \[Pi])/11 - I (b + x)) (-1 - x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (1 + x),
E^((I x)/2) (-1 + x), 1 - x, 1 + x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0,
0, E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0, 0, 0, 0, 0,
0, -1 - x, -1 + x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, -E^(I b - (10 I d \[Pi])/11 - I x) (-1 + x),
E^(-(10/11) I d \[Pi] + I (b + x)) (1 + x), -E^(-((I x)/2)) (1 + x),
E^((I x)/2) (-1 + x), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0,
0, -E^(I b - (10 I d \[Pi])/11 - I x) (1 + x),
E^(-(10/11) I d \[Pi] + I (b + x)) (-1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0, 1 + x, 1 - x,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, -1 + x, -1 - x, -E^(-((I x)/2)) (-1 + x), E^((I x)/2) (1 + x), 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x),
E^((12 I d \[Pi])/11 + I (-b + x)) (1 + x),
E^((12 I d \[Pi])/11 - I (b + x)) (1 - x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((12 I d \[Pi])/11 + I (-b + x)) (-1 + x),
E^((12 I d \[Pi])/11 - I (b + x)) (-1 - x), 0,
0, -E^(-((I x)/2)) (-1 + x), E^((I x)/2) (1 + x), 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 - x, 1 + x, 0,
0, -E^(-((I x)/2)) (1 + x), E^((I x)/2) (-1 + x), 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), -1 - x, -1 + x, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, -E^(-((I x)/2)) (1 + x), E^((I x)/2) (-1 + x), 0,
0, -E^(I b - (14 I d \[Pi])/11 - I x) (-1 + x),
E^(-(14/11) I d \[Pi] + I (b + x)) (1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), -E^(
I b - (14 I d \[Pi])/11 - I x) (1 + x),
E^(-(14/11) I d \[Pi] + I (b + x)) (-1 + x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 1 + x, 1 - x, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (-1 + x),
E^((I x)/2) (1 + x), 0, 0, -1 + x, -1 - x, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, -E^(I b - (18 I d \[Pi])/11 - I x) (-1 + x),
E^(-(18/11) I d \[Pi] + I (b + x)) (1 + x), -E^(-((I x)/2)) (1 + x),
E^((I x)/2) (-1 + x), 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0,
0, -E^(I b - (18 I d \[Pi])/11 - I x) (1 + x),
E^(-(18/11) I d \[Pi] + I (b + x)) (-1 + x), 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0, 1 + x, 1 - x,
0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, -1 + x, -1 - x, -E^(-((I x)/2)) (-1 + x), E^((I x)/2) (1 + x), 0,
0, 0, 0, 0, 0}, {0, 0, -E^(I (t - x)) (1 + x),
E^(I (t + x)) (-1 + x), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, E^(I (b - x)) (1 - x), E^(I (b + x)) (1 + x)}, {0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0,
0, -E^(I (b - x)) (1 + x), E^(I (b + x)) (-1 + x)}, {0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0, 1 + x,
1 - x}, {0, 0, E^(I (t - x)) (1 - x), E^(I (t + x)) (1 + x), 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1 + x, -1 - x}, {0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0, 0, 0, 0, 0,
E^((16 I d \[Pi])/11 + I (-b + x)) (1 + x),
E^((16 I d \[Pi])/11 - I (b + x)) (1 - x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (-1 + x),
E^((I x)/2) (1 + x), E^((16 I d \[Pi])/11 + I (-b + x)) (-1 + x),
E^((16 I d \[Pi])/11 - I (b + x)) (-1 - x), 0, 0, 0, 0, 0, 0, 0, 0,
0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (1 + x),
E^((I x)/2) (-1 + x), 1 - x, 1 + x, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0, 0, 0, 0,
0, 0, -1 - x, -1 + x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0,
E^((I x)/2) (-1 + x), -E^(-((I x)/2)) (1 + x), 0, 0,
E^((20 I d \[Pi])/11 + I (-b + x)) (1 + x),
E^((20 I d \[Pi])/11 - I (b + x)) (1 - x), 0, 0}, {0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (-1 + x),
E^((I x)/2) (1 + x), E^((20 I d \[Pi])/11 + I (-b + x)) (-1 + x),
E^((20 I d \[Pi])/11 - I (b + x)) (-1 - x), 0, 0}, {0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -E^(-((I x)/2)) (1 + x),
E^((I x)/2) (-1 + x), 1 - x, 1 + x, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, E^((I x)/2) (1 + x), -E^(-((I x)/2)) (-1 + x), 0,
0, -1 - x, -1 + x, 0, 0}}
lang-mma