# 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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### Diagonalizing eigensystem to find normal modes of coupled oscillator [closed]

I've two equations of motion that arose in a coupled oscillators in a magnetic field $\rightarrow$ continuum problem in classical mechanics: \begin{eqnarray*} -\omega^2 X &=& - \omega_0^2 X (2 ...
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### Linear algebra as a gauge theory

Is linear algebra a gauge theory? Is the gauge transformation a change of basis? This was the explanation that I received: "Take the principal bundle to be the frame bundle $LM$ of your space $M$...
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### What are the Lorentz Transformations between polar coordinates? Or can Lorentz Transformations be Non-Linear?

This question rises from the comments on @G Smith's answer's to this question https://physics.stackexchange.com/a/603032/113699 Precisely I was trying to understand the Lorentz Transformations between ...
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### Examples of the physical significance and importance of matrix diagonalization and eigenvalues for first year undergraduates? [closed]

To a student of physics, who is only exposed to the techniques of mathematical physics and read classical mechanics at the undergraduate level, but not quantum mechanics yet, how can we explain the ...
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### Normalizing eigenvectors [closed]

Over the course of this quantum class I'm taking I've run into issues with properly normalizing my eigenvectors. Here is my TA's explanation of this particular example is done. I am lost as to where ...
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### Meaning of The Eigenvalues and Eigenvectors of a Quantum Operator

This is more a check to ensure I know the physical meaning of eigenvectors and eigenvalues in quantum mechanics, and to ask the general community if this is wrong: On some observable, represented by ...
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### Constant curvature for maximally symmetric spaces

I'm working through Walds GR Textbook and while reading chapter 5 I stumbled upon the question Proving constant curvature. However, my question is how do we prove that $L$ is symmetric? It is ...
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### Derivation of Cayley-Klein parameters in Goldstein

In his derivation for Cayley-Klein parameters, Goldstein introduces the matrix $$\mathbf{Q}=\begin{pmatrix}\alpha & \beta \\ \gamma & \delta \end{pmatrix}$$ and says that the following unitary ...
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### What's the difference between a linear operator and a tensor?

In Quantum Mechanics we mostly use linear operators which often represent a physical observable, In relativity we mostly work with tensors to describe how things change. But both objects are (multi)-...
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### Does the first-order energy correction in the degenerate case equals to the eigenvalues of the perturbation matrix?

According to Griffiths, the degenerate perturbation theory says that the first-order corrections to the energies are the eigenvalues of the perturbation matrix. Griffiths solves for the eigenvalues in ...
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### Angular change with camera reference point/vector change

Note: All measurements are taken/derived from 2D coordinates based on the video camera. Arm is a shorthand for angle between the hip to the armpit to the elbow. For all intents and purposes, arm can ...
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### Is the Pusey-Barrett-Rudolph (PBR) theorem only applicable in two dimensions?

In the Pusey-Barrett-Rudolph (PBR) paper, https://arxiv.org/abs/1111.3328 it is shown that two non-colinear pure states $|\psi_0\rangle$ and $|\psi_1\rangle$ cannot have overlapping ontic supports (...
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### Dirac Notation and Coordinate transformation of a function

Lets say we have a 1 dimensional system with coordinate $x$ and the associated operator $\hat x$ with eigenstates $|x\rangle$. A function of $x$ is defined as $$f(x) = \langle x |f\rangle \tag{1}$$ ...
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### Why don't the specificities of quantum mechanics (like the necessity of complex number) appear in classical mechanics?

It's well known that classical mechanics is a crude approximation of reality, and that it can be derived from quantum mechanics. But if this is so, why is it not a linear theory, like quantum ...
Consider the following relations $$H_0|\psi_a\rangle = E_a|\psi_a\rangle$$ $$H_0|\psi_b\rangle = E_b|\psi_b\rangle$$ I am struggling then to understand why the following identity holds (its probably ...
I'm working on the Eq.9.77 in Peskin (page 304): To demonstrate this, we need only apply standard identities from linear algebra. First notice that, if a matrix $B$ has eigenvalues $b_i ,$ we can ...