# 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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### Basis of the Lie Algebra of a Group [migrated]

It is known that the Lie algebra of a group is a vector space. The question i have is this: Is there a way to find a basis of the Lie algebra of the group? Also, if i have a set of matrices that ...
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### Generalization to Rear-Wheel Steering?

Given the position of the vehicle (𝑥,𝑦) at different time points, the speed of the vehicle (m/s), the direction the vehicle is facing (heading — in degrees), the track width of the vehicle, and the ...
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### Photon near a black hole - find distance of closest approach from impact parameter

I have the equation relating the impact parameter $b$ to the distance of closest approach $R$. $R^3 - b^2R + 1 = 0$ which can be solved in python. I have a given $b$ and have to find $R$. however, ...
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### Taking out % contribution of a zero order state from an eigenvector - dipole calculation

I am doing an analysis of a theoretical spectroscopy calculation. I take an eigenvector (nx1) and dot it with many zero-order dipole vectors (nx3) to get the dipole contribution to my new eigenstate. ...
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### Can an eigenvalue be a function?

When we say that $$\hat{E}(\psi(x))=\alpha\psi(x),$$ where $\hat{E}$ is an operator and $\alpha$ is the eigenvalue. Is $\alpha$ a fixed constant(like a number) or can it's value keep on varying? ...
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### Uniqueness of simultaneous eigenstates of two linear operators

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigenstates ...
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### Infinite Coupled Masses, symmetry, and the simultaneous diagonal theorem for infinite dimensional vector spaces

In The Physics of Waves by Georgi, in Chapter 4, we show that, in a coupled system of masses connected by springs, a transformation that preserves some symmetry $S$ commutes with $K^{-1}M$. From my ...
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### Why do the density operators span the whole operator space $\mathcal{B}(H)$?

The convex set of density operators on a finite-dimensional Hilbert space $H$ defined by $$\mathcal{D}(H):=\{\rho\in\mathcal{B}(H)\,|\,\rho\geq 0,\, \operatorname{tr}\rho =1\},$$ This set is said to ...
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### How does a linear operator act on a bra?

I'm studying QM from Shankar. He introduces linear operators and says that an operator is an instruction for transforming one ket into another. But then a few lines below he says operators can also ...
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### Why is the dimension of the set separable states $\dim\mathcal H_1+\dim\mathcal H_2$?

Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$? This is the relevant passage: So far, we have assumed implicitly that the ...
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### Complex conjugate and transpose “with respect to a basis”

In my quantum mechanics notes, my teacher described the complex conjugate and transpose of a linear operator X as "with respect to an orthogonal basis." What does it mean to take a transpose or ...
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### Determine the point at which moment vector is zero on a 3D body

I have information about total force and moment on a body for three points, whose coordinates I know. From this information I would like to determine the point at which moment would be zero. ...
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### Vectors ( Resolution of vectors )

How many components can a vector be resolved into? I think that it should be infinity because there can be infinite axes. Am I right?
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### Proof of skewsymmetry of electromagntic function in Minkowski spacetime

I have been studying special relativity from the Gregory Naber's book: "The geometry of Minkowski spacetime" and I found a very strange proof. In Section 2.1, just before of equation 2.1.2. the book ...
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### Fermionic statictics in $SU(2)$ slave-boson representation

One of the $SU(2)$ slave-boson decompositions has been introduced by X.-G. Wen and P. A. Lee in PRL, 76, 503 (1996). (A generic recipe for constructing the SU(2) slave-particle framework has been ...
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### Moment of Inertia Tensor Terminology

I've learned about the moment of inertia tensor as a matrix that can be used to compute angular momentum, moment of inertia, etc. for a system. But why is it often described as a tensor instead of a ...
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### Why are physicists so interested in irreps if in their non-block-diagonal form they mix all components of a vector?

Consider a group $\{G,\circ\}$, with elements $e,g_1,g_2,...$, represented by the matrices $\{D(e), D(g_1), D(g_2)...\}$. If all the matrices can be brought to block diagonal forms by a similarity ...
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### What's the physical meaning of the kernel of density matrix?

The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector. But what's the physical meaning of the kernel of density matrix?
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### Order of positions of tensor/vector components in an inner/outer product

Show that if $T_i$ are the components of covariant vector T, then $S_{ij}=T_iT_j-T_jT_i$ are the components of a skew-symmetric covariant tensor S. The question is whenever working with equations of ...
When we solve for inner product of $\rvert a \rangle \cdot \rvert b \rangle$ we solve for $\langle a \rvert b \rangle$ where $\langle a \rvert$ is complex conjugate of $\rvert a \rangle$. However this ...