# Questions tagged [linear-algebra]

To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.

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### Feynman diagrams for eigenvalue perturbation theory

I posted this question in MathOverflow but was not lucky with the answers, so wil try here. Suppose I have a matrix given by a sum $$A=D+\epsilon B$$ where $D$ is diagonal and $\epsilon$ is small, ...
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### What is the role of determinant and trace of matrices in physics? [closed]

There is vast area of physics where we have to use matrices.It is not only to do the mathematical problems in physics but also to produce a physical realization of an operation. I think matrices carry ...
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### Charge conjugation in chiral representation

I'm reading Maggiore's book and I got to the part of charge conjugation symmetry for Dirac spinor. I get that the definition of charge conjugation is representation-dependent, however I couldn't find ...
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### Property of the adjoint operator in the array element

In Quantum Mechanics how can I prove this property? $$<\psi|A^{\dagger} |\phi>=<\phi|A|\psi>^{*}$$
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### Demonstration of a property of the adjoint operator in Quantum Mechanics

I would like to demonstrate the following property: $$(|\psi\rangle\langle\phi|)^{\dagger}=|\phi\rangle\langle\psi|$$ The other properties that concern the adjoint operator I have already been ...
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### State of a linear vector space

I was going through a lecture on linear vector space. It embodied all the examples of a linear vector space like it could be function spaces or vectors or real entities. But what exactly is meant by ...
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### How does a linear operator act on vectors that are not eigenvectors of the operator?

I feel like I have the wrong end of the stick in thinking about quantum state vectors, and so wish to try and clear this up. If I prepare a single spin in the up state (along the z-dimension), and ...
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### Commutativity vs Compatibility

As far as I know, two compatible observables have a complete set of common eigenvectors, and using this fact, one can prove that their corresponding operators are commutative. Well now is the converse ...
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### How do tensors point in multiple directions?

I have been learning about tensors recently but I am still confused about one thing. I understand that tensors are just objects whose elements transform in a fixed way so that the object is coordinate ...
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### Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
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### Some beginner doubt about tensors

I have a question about tensors. Okay, you could say that this question would fits more properly on Math.StackExchange but, here it's more cozy. Anyway: A vector, as we all learned in first year, ...
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### Proving that unitary transformations preserve orthonormality in bracket notation

I'm trying to prove that unitary transformations preserve orthonormality for quantum chemistry course. There is a proof for this in many mathematical texbooks, but non of them is dealing with matrix ...
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### Inverse of a Hermitian Operator

The momentum operator $p = i\frac{\mathrm d}{\mathrm dx}$ (with $\hbar = 1$) is hermitian. Hence its imaginary exponential i.e. $U = e^{ip}$ must be unitary. $U$ being a unitary operator, must have a ...
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### What eigenvector-like tools are there for analyzing tensors of rank three and higher?

If I have a rank-two tensor that I want to analyze ─ say, an electric quadrupole moment, or a moment of inertia ─ it can often be very easy to analyze by moving to its principal-axes frame: one ...
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### Finding correct zeroth order eigenstates in degenerate perturbation theory when degeneracy is not lifted by first order correction

I have recently learned that if the degeneracy is not lifted in the first order in degenerate perturbation theory, one has to diagonalize a different matrix that plays a similar role to the first ...
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### How does the addition of two wavefunctions develop in time?

Two time dependent wavefunctions: $\Psi _1(t)= \psi_1*exp(\frac{-i * E_1}{\hbar}*t)$ $\Psi _2(t)= \psi_2*exp(\frac{-i * E_2}{\hbar}*t)$ Both a solution to the timeindependent (note "in") ...
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### Minimum uncertainty Gaussians as basis for Hilbert space?

In a class on QM the lecturer (near min. 47) briefly says that gaussians of minimum uncertainty can form a basis for Hilbert space, meaning that any element of the space can be expressed as a linear ...
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### How can I compute the eigenstates of a tight-binding Hamiltonian describing a system?

I have the following set-up in a 3-site tight-binding system: \begin{align} i\hbar\frac{dc_1}{dt}&=-Ac_2,\\ i\hbar\frac{dc_2}{dt}&=-Ac_1-Ac_3,\\ i\hbar\frac{dc_3}{dt}&=-Ac_2, \end{align} ...
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### Eigenvalues of the Klein-Gordon operator

If I've understood what I've read correctly, the eigenvalues of the Klein-Gordon (KG) operator $\Box+m^{2}$ are $-p^{2}+m^{2}$, but how does one show this? Naively I assumed that the eigenfunctions ...
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### Significance of using Eigenvalues / Vectors in QM?

A fundamental idea in Quantum Mechanics is that observable quantities are represented by linear, hermitian operators. Why is it that we represent distinguishable states as the eigenvectors of ...
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### Product of operators, eigenvalues

If I've two hermitian operators let's say A and B,then their eigenfunctions(vectors) form a basis... If I now take a product of both of them and create a new operator AB (composition of both), that ...
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### How is the Gastner-Newman equation implemented to create value-by-area cartograms? [closed]

There is a paper called "Density-equalizing map projections: Diffusion-based algorithm and applications" by Michael T. Gastner and M. E. J. Newman, which explains their algorithm (which is based in ...
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### Transformation of a covariant tensor and general interpretation

If we consider a coordinate transformation defined by the rule $$x^{\mu}\rightarrow x'^{\mu}=x'^{\mu}(x)$$ If we consider the old coordinates as functions of the new ones, then the coordinate ...
I have been asked to prove that, for an Hermitian operator A, its standard deviation $\Delta_{\varphi}$ in a certain state $|\varphi\rangle$ is equal to $0$ if, and only if, said state is an ...