How to write a matrix in a different basis?

Questions
1

Hi!

I have these two 4x4 matrix

Subscript[S, 1] = KroneckerProduct[PauliMatrix[1], IdentityMatrix[2]]

Subscript[S, 2] = KroneckerProduct[IdentityMatrix[2], PauliMatrix[2]]

Which are written in the Canonical basis ({{1}, {0}, {0}, {0}}; {{0}, {1}, {0}, {0}}; {{0}, {0}, {1}, {0}}; {{0}, {0}, {0}, {1}})

I would like to write down these two matrices in the following basis:

G := {{0}, {0}, {0}, {1}}

S := (1/Sqrt[2]) {{0}, {1}, {1}, {0}}

A := (1/Sqrt[2]) {{0}, {1}, {-1}, {0}}

E := {{1}, {0}, {0}, {0}}

Is there any way to do it in Mathematica?

Thanks in advance

2

According to my books this should be done by

m1 = KroneckerProduct[PauliMatrix[1], IdentityMatrix[2]];
mT = {
   {0, 0, 0, 1},
   {0, 1/Sqrt[2], 1/Sqrt[2], 0},
   {0, 1/Sqrt[2], -1/Sqrt[2], 0},
   {1, 0, 0, 0}};


m1T = Inverse[mT].m1.mT;
m1T // MatrixForm
3

Thanks for the answer!

I understood what you have done but when I reply it I do not get a matrix as the result. Instead, I get the following result:

Capturadetela2021-07-21201523.png
Capturadetela2021-07-21201523.png1347×117 9.44 KB

4

I cannot understand what is going on there. It works on my system. Did you copy/paste or did you type the commands?

5

Sorry! The problem was my code. It works here now. Thank you very much!

6

Crossposted here.