Proving Hermition Matrix Real & Eigenvalues/Eigenspinors

[eman2009]

in the spin matrix if the <A> is real how we can prove the matrix is herimition ,and how we can prove the eigenvalues of any hermition 2x2 matrix is real and its eigenspinors are orthognal if the eigenvalues is defferent and if it is the same? and A physical quantity

 

[Meir Achuz]

Look in a math physics book.

 

[Entropia]

To prove that a matrix is Hermitian, we need to show that it is equal to its own conjugate transpose. In other words, the matrix A is Hermitian if A = A†, where A† is the conjugate transpose of A. If the spin matrix <A> is real, then A† = A and we can easily see that A = A†.

To prove that the eigenvalues of a Hermitian 2x2 matrix are real, we can use the fact that the eigenvalues of a Hermitian matrix are always real. This means that if a matrix is Hermitian, its eigenvalues must also be real. In the case of a 2x2 matrix, we can use the characteristic polynomial to find the eigenvalues, and since the coefficients of the polynomial are real, the eigenvalues must also be real.

To prove that the eigenspinors of a Hermitian matrix are orthogonal, we can use the fact that the eigenvectors of a Hermitian matrix are always orthogonal. This means that if a matrix is Hermitian, its eigenvectors must also be orthogonal. In the case of a 2x2 matrix, we can use the eigendecomposition to find the eigenvectors, and since the eigenvectors are orthogonal, the eigenspinors must also be orthogonal.

A physical quantity can be represented by a Hermitian matrix in quantum mechanics. This matrix represents the observable associated with the physical quantity, and its eigenvalues represent the possible outcomes of a measurement of that observable. The eigenspinors represent the states in which the observable has a definite value, and their orthogonality ensures that the probabilities of obtaining different outcomes are well-defined. Therefore, proving the Hermiticity, real eigenvalues, and orthogonal eigenspinors of a matrix is crucial in understanding and predicting the behavior of physical systems in quantum mechanics.