# Today I Learned

## Interesting Facts on Eigenstates

The eigenstates of a matrix

are

However if we put in two real off-diagonal elements

the eigenstates becomes

If we are talking about real matrix,

even though it is true that the eigenstates are

the eigenstates are not that simple if we add in off-diagonal elements that make sure the matrix is still Hermitian.

In fact the eigenstates for the eigenequation

is given by

where $b^2 = b \cdot b$ is not the modulus squared.