What is Hermitian: Definition and 350 Discussions

Numerous things are named after the French mathematician Charles Hermite (1822–1901):

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1. I Generalized Eigenvalues of Pauli Matrices

Consider a generic Hermitian 2x2 matrix ##H = aI+b\sigma_{x}+c\sigma_{y}+d\sigma_{z}## where ##a,b,c,d## are real numbers, ##I## is the identity matrix and ##\sigma_{i}## are the 2x2 Pauli Matrices. We know that the eigenvalues for ##H## is ##d\pm\sqrt{a^2+b^2+c^2}## but now suppose I have the...
2. A Can a Non-Diagonal Hermitian Matrix be Diagonalized Using Unitary Matrix?

Every hermitian matrix is unitary diagonalizable. My question is it possible in some particular case to take hermitian matrix ##A## that is not diagonal and diagonalize it UAU=D but if ##U## is not matrix that consists of eigenvectors of matrix ##A##. ##D## is diagonal matrix.
3. Show that an operator is Hermitian

Hi, unfortunately, I have problems with the following task I tried the fast way, unfortunately I have problems with it I have already proved the following properties, ##\bigl< f,xg \bigr>=\bigl< xf,g \bigr>## and ##\bigl< f, \frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g...
4. POTW Prove Hermitian Matrices Satisfy ##H^2 = H^\dagger H##

Show that an ##n\times n##-matrix ##H## is hermitian if and only if ##H^2 = H^\dagger H##.
5. Is an operator (integral) Hermitian?

Knowing that to be Hermitian an operator ##\hat{Q} = \hat{Q}^{\dagger}##. Thus, I'm trying to prove that ##<f|\hat{Q}|g> = <\hat{Q}f|g> ##. However, I don't really know what to do with this expression. ##<f|\hat{Q}g> = \int_{-\infty}^{\infty} [f(x)^* \int_{-\infty}^{\infty} |x> <x| dx f(x)] dx##...
6. I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...
7. I Hermitian Inverse: Exploring Eigenvalues of ##H^{-1}##

##H## is an ##n\times n## Hermitian matrix with eigenvectors ##\mathbf{e}_i## and all eigenvalues negative. It's claimed that ##G = \int_{0}^{\infty} e^{tH} dt## is such that ##G = H^{-1}##. I was looking at\begin{align*} G\mathbf{e}_i &= \int_0^{\infty} \sum_{n=1}^{\infty} \frac{t^n}{n!} H^n...
8. A Finding the Hermitian generator of a Symplectic transformation

Consider a set of ##n## position operators and ##n## momentum operator such that $$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$ Lets now perform a linear symplectic transformation $$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$ $$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$ such that the canonical commutation...
9. I A strange definition for Hermitian operator

In lecture notes at a university (I'd rather not say which university) the following definition for Hermitian is given: An operator is Hermitian if and only if it has real eigenvalues. I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We...
10. B Is the Momentum Operator Hermitian? A Proof

Momentum operator is ##p=-i\frac{d}{dx}## and its adjoint is ##p^\dagger=i\frac{d}{dx}##. So, ##p^\dagger=-p##. How is the momentum Hermitian?
11. I General worked out solution for diagonalizing a 4x4 Hermitian matrix

Hello, I am looking for a worked out solution to diagonalize a general 4x4 Hermitian matrix. Is there any book or course where the calculation is performed? If not, does this exist for the particular case of a traceless matrix? Thank you!
12. A How can we measure these Hermitian operators?

Hi Pf, I am reading this article about generalization of Pauli matrices https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices#Generalized_Gell-Mann_matrices_%28Hermitian%29 When i receive a qubit in a given density matrix , i can measure the mean values of the Pauli matrices by...
13. I Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

The basis he is talking about: {1,x,x²,x³,...} I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the...
14. I Anti-unitary operators and the Hermitian conjugate

The definition of the hermitian conjugate of an anti-linear operator B in physics QM notation is \langle \phi | (B^{\dagger} | \psi \rangle ) = \langle \psi | (B | \phi \rangle ) where the operators act to the right, since for anti-linear operators ( \langle \psi |B) | \phi \rangle \neq...
15. Understanding Commutativity and Eigenvalues in the Product of Hermitian Matrices

Product of two Hermitian matrix ##A## and ##B## is Hermitian matrix only if matrices commute ##[A,B]=0##. If that is not a case matrix ##C=AB## could have complex eigenvalues. If A=\sum_k \lambda_k|k \rangle \langle k| B=\sum_l \lambda_l|l \rangle \langle l| AB=\sum_{k,l}\lambda_k\lambda_l|k...
16. A Converting between field operators and harmonic oscillators

Suppose we have a Hamiltonian containing a term of the form where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use where I have set ħ=1 so that This term is Hermitian because r and p both are.*...
17. Hermitian Matrix and Commutation relations

I think I roughly see what's happening here. > First, I will assume that AB - BA = C, without the complex number. >Matrix AB equals the transpose of BA. (AB = (BA)t) >Because AB = (BA)t, or because of the cyclic property of matrix multiplication, the diagonals of AB equals the diagonals of...

49. I Hermitian Operators in Dirac Equation

In the dirac equation we have a term which is proportional to \alpha p . In the book they say that \alpha must be an hermitian operator in order for the Hamiltonian to be hermitian. As I understand, we require this because we want (\alpha p)^\dagger = \alpha p. But (\alpha p)^\dagger =...
50. I From Non Hermitian to Hermitian Matrix

Is there any way that i can convert a non-hermitian matrix to a hermitian matrix ?