2024-01-31 22:44:35 +0200 edited question Solving a linear system of equations depending of a parameter Solving a linear system of equations depending of a parameter Hi, Let $a$ be a fixed parameter, and let $$(S) : \begin{ 2024-01-31 22:44:13 +0200 edited question Solving a linear system of equations depending of a parameter Solving a linear system of equations depending of a parameter Hi, Let a be a fixed parameter, and let [(S) : \begin{c 2024-01-31 22:43:57 +0200 edited question Solving a linear system of equations depending of a parameter Solving a linear system of equations depending of a parameter Hi, Let a be a fixed parameter, and let$$(S) : \begin{ 2024-01-31 22:43:05 +0200 edited question Solving a linear system of equations depending of a parameter Solving a linear system of equations depending of a parameter Hi, Let $a$ be a fixed parameter, and let $(S) : \begin{c 2024-01-31 22:42:46 +0200 edited question Solving a linear system of equations depending of a parameter Solving a linear system of equations depending of a parameter Hi, Let$a$be a fixed parameter, and let$(S) : \begin{c 2024-01-31 22:42:12 +0200 asked a question Solving a linear system of equations depending of a parameter Solving a linear system of equations depending of a parameter Hi, Let $a$ be a fixed parameter, and let \$(S) : \begin{c 2023-10-25 15:00:26 +0200 commented answer Simplify a matrix with symbolic variables Thank you ! 2023-10-25 15:00:00 +0200 marked best answer Simplify a matrix with symbolic variables Hello, I've tried the following code in SageMath, but I don't succeed in getting a simplified expression (even using the method .simplify_full()) : var('w_2, w_3, w_4', domain='complex') a_2=sqrt(1-norm(w_2))*sqrt(1-norm(w_3)) a_3=sqrt(1-norm(w_3))*sqrt(1-norm(w_4)) b_2=-conjugate(w_3)*sqrt(1-norm(w_2))*sqrt(1-norm(w_4)) C=Matrix([[0, w_2, a_2, b_2], [0,0,w_3,a_3], [0,0,0,w_4], [0,0,0,0]]) Id=matrix.identity(4) T=Id-C*(C.H) T.simplify_full()  I get the "awfull" outcome [-sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*conjugate(sqrt(-w_2*conjugate(w_2) + 1))*conjugate(sqrt(-w_4*conjugate(w_4) + 1))*conjugate(w_3) - sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*conjugate(sqrt(-w_2*conjugate(w_2) + 1))*conjugate(sqrt(-w_3*conjugate(w_3) + 1)) - w_2*conjugate(w_2) + 1 sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(sqrt(-w_3*conjugate(w_3) + 1))*conjugate(sqrt(-w_4*conjugate(w_4) + 1))*conjugate(w_3) - sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_3*conjugate(w_3) + 1)*conjugate(w_3) sqrt(-w_2*conjugate(w_2) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_3)*conjugate(w_4) 0] [ sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*w_3*conjugate(sqrt(-w_2*conjugate(w_2) + 1))*conjugate(sqrt(-w_4*conjugate(w_4) + 1)) - w_3*conjugate(sqrt(-w_2*conjugate(w_2) + 1))*conjugate(sqrt(-w_3*conjugate(w_3) + 1)) -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(sqrt(-w_3*conjugate(w_3) + 1))*conjugate(sqrt(-w_4*conjugate(w_4) + 1)) - w_3*conjugate(w_3) + 1 -sqrt(-w_3*conjugate(w_3) + 1)*sqrt(-w_4*conjugate(w_4) + 1)*conjugate(w_4) 0] [ w_3*w_4*conjugate(sqrt(-w_2*conjugate(w_2) + 1))*conjugate(sqrt(-w_4*conjugate(w_4) + 1)) -w_4*conjugate(sqrt(-w_3*conjugate(w_3) + 1))*conjugate(sqrt(-w_4*conjugate(w_4) + 1)) -w_4*conjugate(w_4) + 1 0] [ 0 0 0 1]  whereas when we compute this "by hand", we can show that T= [[norm(w_3*w_4)*(1-norm(w_2)), -conjugate(w_3)*sqrt(1-norm(w_2))*sqrt(1-norm(w_3))*norm(w_4), conjugate(w_3*w_4)*sqrt(1-norm(w_2))*sqrt(1-norm(w_4))], [-w_3*sqrt(1-norm(w_2))*sqrt(1-norm(w_3))*norm(w_4), norm(w_4)*(1-norm(w_3)), -conjugate(w_4)*sqrt(1-norm(w_3))*sqrt(1-norm(w_4))], [w_3*w_4*sqrt(1-norm(w_2))*sqrt(1-norm(w_4)), -w_4*sqrt(1-norm(w_3))*sqrt(1-norm(w_4)), 1-norm(w_4)]]  Thus, when I want to diagonalize T using the T.eigenvalues() and T.eigenvectors_right(), I get something really complicated... whereas we can show "by hand" that the eigenvalues are just 0 and 1-norm(w_2*w_3*w_4)... How could we manage to get those simple expressions with Sage Math ? Thanks in advance for your help ! 2023-10-25 14:59:47 +0200 commented question Simplify a matrix with symbolic variables Of course... I've forgotten the last row (but this row is quite trivial...). 2023-10-19 16:27:33 +0200 edited question Simplify a matrix with symbolic variables Simplify a matrix with symbolic variables Hello, I've tried the following code in SageMath, but I don't succeed in gett 2023-10-19 16:27:00 +0200 asked a question Simplify a matrix with symbolic variables Simplify a matrix with symbolic variables Hello, I've tried the following code in SageMath, but I don't succeed in gett 2023-10-19 13:56:35 +0200 commented answer How to define *complex* symbolic variables Thank you :) 2023-10-19 13:56:15 +0200 marked best answer How to define *complex* symbolic variables Hello, I am quite new with SageMath, and I don't succeed in defining complex symbolic variables. In the following example, SageMath seems to compute as if the variables were real (and not complex) : var('w_2') T=Matrix([[0, w_2], [0,0]]) S=T*(T.H) S.eigenvalues()  The output of this command is [w_2^2, 0]  whereas the expected result was [norm(w_2), 0]  How can I tell SageMath that my symbolic variables are complex ones ? Thanks in advance for your help ! 2023-10-19 13:56:15 +0200 received badge ● Scholar (source) 2023-10-19 10:49:10 +0200 received badge ● Editor (source) 2023-10-19 10:49:10 +0200 edited question How to define *complex* symbolic variables How to define *complex* symbolic variables Hello, I am quite new with SageMath, and I don't succeed in defining complex 2023-10-19 10:13:09 +0200 asked a question How to define *complex* symbolic variables How to define *complex* symbolic variables Hello, I am quite new with SageMath, and I don't succeed in defining complex