How many real points are there on the unit circle?debjit625 said:
so that means I have no idea ,now that states my problem very well.debjit625 said:
what you are taking about I have no idea I am in first year and we just started learning these..BvU said:
I guess infinite but what I meant was that they are all at a distance 1 unit from the center.PeroK said:
perhaps gave us the wrong impression. Do you know what unit circle ?debjit625 said:
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the elements on the main diagonal are real numbers, and the elements below the main diagonal are the complex conjugates of the elements above the main diagonal.
A unitary matrix is a square matrix whose conjugate transpose is equal to its inverse. This means that multiplying a unitary matrix by its conjugate transpose will result in the identity matrix.
To show that all eigenvalues of a Hermitian matrix are ±1, we can use the spectral theorem, which states that a Hermitian matrix can be diagonalized by a unitary matrix. This means that the eigenvalues of a Hermitian matrix will be the diagonal entries of the diagonalized matrix, and since a unitary matrix has eigenvalues of magnitude 1, the eigenvalues of the diagonalized matrix (and therefore the original Hermitian matrix) will also have magnitude 1.
Having eigenvalues of ±1 for a Hermitian matrix is important because it allows us to easily determine the inverse of the matrix. Since the eigenvalues are either 1 or -1, their reciprocals are also 1 or -1, making the inverse of the matrix relatively simple to calculate.
Yes, it is possible for a matrix to have eigenvalues of ±1 without being both Hermitian and unitary. For example, a diagonal matrix with diagonal entries of ±1 will have eigenvalues of ±1, but it is not necessarily Hermitian or unitary.