Hermitian Symmetric Matrices

[SLD]

HSM's are complex matrices where the transpose is the complex conjugate of the matrix. All of the diagonal elements are real. For example:

1 i
-i 0

All of their eigenvalues and eigenvectors are real. Pauli came up with three of them for use in quantum physics.

Aside from quantum mechanics, and indeed aside from the Pauli Matrices, what other applications are there for such matrices? Can they be used to create some strange geometries? Computer graphics?

SLD

 

[barbos]

Pauli introduced spinors into QM and corresponding matrices. All QM operators corresponding to measurables (Energy, momentum and angular momentum) are hermitian operators, not just spin.
Dot product of vectors is hermitian, <v1|v2>=<v2|v1>*, so here is your use in geometry
And hermitian operators are generators of unitary operators where U=exp(i*H)
Unitary operators preserve inner product, they are effectively geometric rotations
What prompted this question? Did you just learn about them?

 

[SLD]

Pauli introduced spinors into QM and corresponding matrices. All QM operators corresponding to measurables (Energy, momentum and angular momentum) are hermitian operators, not just spin.
Dot product of vectors is hermitian, <v1|v2>=<v2|v1>*, so here is your use in geometry
And hermitian operators are generators of unitary operators where U=exp(i*H)
Unitary operators preserve inner product, they are effectively geometric rotations
What prompted this question? Did you just learn about them?

Reading a book on quantum physics. They explained the hermitian and the Pauli matrices.

I was wondering though about graphical applications of them and can’t find much about them. Do they do anything maps of complex functions for example? We didn’t study them when I took complex variables 30 years ago.

SLD

 

[barbos]

I am afraid it does not have much applications in graphics. Hermitian is just a particular property of an operator