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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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Sling loads for multipoint lifts

I am trying to calculate sling loads for a n-point lift. I want to utilize vector calculations and make it as general as possible, and also work in 3D-space. The idea is to use position vectors for ...
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1answer
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Prove acceleration in orbit with newtons second law

I want to prove that the acceleration in a orbit at a given point r=(x,y) is $a=-\frac{GM}{R^3}r$ (My professor said this can be proven by newtons second law, but he never explained in detail how). I ...
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1answer
38 views

Question on notation for the inner product of complex vectors

Regarding the wiki: https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form you can see that the wiki states that physics defines the inner product for complex vectors as: $$\langle \, \...
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Generalisation of finite to infinite vector space [closed]

To generalise finite vector space into infinite vector space..this book(principle of quantum mechanics) says we redifine inner product of 〈f|g〉as [ limit n tends to infinity Σfₙ(xᵢ).gₙ(xᵢ)Δ ]where ...
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1answer
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Can someone please explain the meaning of the circled paragraph?

why does the off diognal elements of the matrix mediate with the coupling differential equation?
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52 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
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What is the meaning of “linear” in linear Vector space? [migrated]

$$a\cdot x_1+b\cdot x_2+c\cdot x_3+...+qx_n=\text{constant}$$ is called a linear equation because it represents the equation of a line in an $n$ dimensional space. So "linear" comes from the word "...
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Formal Definition of Dot Product

In most textbooks, dot product between two vectors is defined as: $$\langle x_1,x_2,x_3\rangle \cdot \langle y_1,y_2,y_3\rangle = x_1 y_1 + x_2 y_2 + x_3 y _3$$ I understand how this definition ...
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Normal mode decomposition of a triangular hexagonal lattice

I was trying to understand and redo the methods used in a previous question: Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice ...
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1answer
60 views

Why are coherent states not linearly independent?

From the completeness relation one can see that, $$|\psi \rangle = \int \frac{d^2 \alpha}\pi \langle \alpha | \psi \rangle |\alpha\rangle.$$ And if $|\psi\rangle = |\beta \rangle$ (which is another ...
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Math of anyons: Quantum dimension of 1 implies abelian charge

This question originates from the following statement in Bonderson's thesis: Link to Thesis page 16 or pdf-page 23: The quantum dimension $d_a$ of an anyon of charge $a$ satisfies $d_a \geq 1$ with ...
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Matrix multiplication of $n$-dimensional matrices [migrated]

While reading matrix algebra, I came across the index notation. The multiplication of two dimensional matrices could be written as $$(A_{ij})(B_{jk})=(C_{ik}).$$ How do we generalise this notation for ...
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1answer
51 views

Linear and angular speeds of a train

I am using Radar sensor in a train. Sensor is in the front part of the train to detect object and avoid collision. It needs vehicle motion data to calculate objects longitudional and lateral speeds....
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How to understand notation in “Introduction to Quantum Mechanics (3rd Edition)” by David Griffiths, Chapter 3.6.2?

In the 3rd edition, on page 118, the projection operator is introduced as $$\hat{P}=|\alpha\rangle\langle\alpha|.$$ Then Griffiths says that when $\hat{P}$ acts on another vector, it looks like ...
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1answer
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How can quantum operators be expressed as a matrices?

I have just started quantum mechanics with Shankar. In my understanding, quantum operators are linear operators in infinite-dimensional Hilbert spaces. Shankar has repeatedly treated quantum ...
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23 views

How contravarient metric tensor equals the cofactor of covarient metric tensor over its determinant? [closed]

Is the contravariant form of the metric tensor the inverse of the covariant form of the metric tensor?
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1answer
36 views

Microcausality when quantizing the real scalar field with anticommutators

We know by the spin-statistics theorem that the real scalar field has to be canonically quantized by commutators. But if we try to use anticommutators, we would expand the field $$\phi(x)=\int\frac{d^...
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2answers
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Exact solution for the perturbation of the inverse metric

So when we usually linearize general relativity with respect to metric perturbations $g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}$, we compute the correction to the inverse of the metric to first ...
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0-rank tensor vs vector in 1D

What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D? As far as I understand tensor is anything which can be measured and different measures can be transformed into ...
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1answer
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Is any given triplet spin state an eigenstate of some $j^z$ in the suitable basis?

Imagine you have a triplet spin state, which, in general, can be written as $$|\psi \rangle = \alpha | \uparrow \uparrow \rangle + \beta ( | \downarrow \uparrow \rangle+ | \uparrow \downarrow \...
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1answer
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Is a dichotomic basis possible for 3-dimensional space?

We know that the Pauli basis for the 2-dimensional space is a dichotomic basis in the sense that every Pauli matrix has two distinct eigenvalues. Is it possible to express a 3-dimensional matrix $\...
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Recovering symmetry in coupled oscillators

Consider a pair of LC oscillators, one with capacitance $C_1$ and inductance $L_1$ and the other with capacitance $C_2$ and inductance $L_2$. Suppose they're connected through a capacitor $C_g$. We ...
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3answers
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Invariance of $ds^2$ from invariance of all null intervals

Is this linear algebra statement true? Let $\eta= \begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}$. If $x^T (\Lambda^T\eta\Lambda) x$...
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2answers
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Infinitesimal Rotation under Orthogonal Similarity Transformation

I’m reading charpter 4.8 of Goldstein’s classical mechanics 3rd edition that deals with infinitesimal rotations, and the following is the part I got stuck: (p.166~167) If $d\boldsymbol{\Omega}$ is ...
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Demonstration of the completness of an orthonormal set of functions

I find this concept of completness a little bit dense when it comes to prove this property of some set of orthonormal functions. In one of my classes, my professor proved this for the orthonormal set ...
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1answer
58 views

Determinant of ADM metric

I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form: $$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
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1answer
102 views

Matrix of Hamiltonian $H=\frac{ħω}{2}(|1〉〈1|−|2〉〈2|)+\frac{iħχ}{2}(|1〉〈2|−|2〉〈1|)$ [closed]

I have a second order system, with a Hamiltonian $$H=\frac{ħω}{2}(|1〉〈1|−|2〉〈2|)+\frac{iħχ}{2}(|1〉〈2|−|2〉〈1|)$$ where $|1〉,|2〉$ form a complete basis for the system. I'm trying to get the matrix that ...
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1answer
59 views

Verifying the “quantum teleportation identity” in $\mathbb{C}^2 \otimes \mathbb{C}^2$ (Bell basis)

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}$ Let $\{\psi_{i,j} : i, j = 0,1\}$ be the Bell basis of $\...
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Simultaneous shifted diagonalization of bunch of operators

I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule $$ \Big[ L_i ,L_j \Big]= (i-j)(L_{i+j}-\frac14 L_{i+j-1}).....*$$ My question is ...
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Why do the matrix elements of an operator correspond to the Fourier components of the observable in Heisenberg's Matrix Mechanics?

It is well-known that Heisenberg $a$ began developing his Matrix Mechanics by creating matrix components $$A(n,n-a,t)=A(n,n-a)e^{i\omega(n,n-a)t}$$ or $$A_{nm}(t)=A_{nm}e^{\frac{i}{\hbar}\omega(nm)t}$$...
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1answer
412 views

Exponential of the Pauli matrices [closed]

My job is to prove: $$\exp(i\theta \vec{v} \cdot \vec{ \sigma })=\cos(\theta)I+i\sin(\theta)\vec{v} \cdot \vec{ \sigma }$$ where $\theta \in \mathbb{R}$ and $\vec{v} \cdot \vec{ \sigma }=\Sigma^3_{i=...
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Proving two space-time intervals are equivalent with matrix algebra

η=$ \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $ v=$ \begin{bmatrix} ct\\ x\\ y\\ z \end{...
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Is Smoluchowski equation for aggregation conservative?

The Smoluchowski equation for the aggregation with no breakage can be written as $$\frac{dn_k}{dt}=\frac{1}{2}\Big(\sum_{i+j=k} n_i \beta_{ij}n_j\Big) - n_k \sum_{i=1}^N \beta_{ki} n_i$$ where $n_k$ ...
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Applying Sylvester's theorem in quantum mechanics

A $2d$ system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is an unitary Hermitian $2$x$2$ matrix with eigenvalues $\exp(±i\theta)$. I need to use ...
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2answers
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What does it mean for a unit vector to have a magnitude of 1?

Imagine a Cartesian coordinate system whose origin is associated with two unit vectors, ê and â, in a 2D-space. Now, let 0.5 cm be the unit of length in this coordinate system. The magnitude of a ...
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2answers
66 views

Unit Vectors in physics

I'm reading the Massachusetts Institute of Technology: "Review of Vectors" , and I've found this: I can't see any relationship between the text that is highlighted in yellow and what's depicted in ...
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2answers
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Rank of a density matix

I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
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Is the basis of the occupation numbers of bosonic system orthonormal?

I am interested in the calculation of a correlation function in the Fock space of a system of $N$ bosons. As a trace it would be convenient for me to sum over the elements of the occupation numbers ...
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2answers
110 views

How does the partial transpose operation look like in matrix form?

The link here gives a nice description of how partial trace looks in matrix notation. I want a similar explanation for the matrix partial-transposition. How does matrix partial-transposition operation ...
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1answer
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Why the basis of vectors and one-forms can not be related through the metric as a vector and one-forms?

I know that basis vector and basis of one-forms are related through $$ \tilde{e}^\mu \cdot \vec{e}_\nu = \delta^\mu _\nu .\tag{1}$$ However, the metric has the property that allows to convert ...
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1answer
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Gram-Schmidt process and degenerate subspace of the solutions to the Schrodinger's equation

So I know that in QM each linear combination of a degenerate set of wavefunctions is also a solution to the Schrodinger's equation (SE). The degenerate wavefunctions must be orthogonal to the non-...
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1answer
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On the dimensionality of the Hilbert space for a central potential [duplicate]

So, we have that for particles in a central potential, the wavefunction can be expressed as: $$\psi(\mathbf{x})=R(r)Y^m_l(\theta,\phi)$$ Explanation (not crucial in the question): My question is: ...
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Why do we get wrong answers for orbital angular momentum if we solve it algebraically?

It is a well-known fact that the values for the square of the orbital angular momentum of a particle $L^2$ and it's projection in the $z$-direction $L_z$ are $m\hbar$ and $l(l+1)\hbar$ and that $l$ ...
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1answer
91 views

Proof of invariance of scalar product under rotations, using index notation

So I have got the following question: Show that the scalar product of two cartesian vectors $p_i\cdot q_i$ is invariant under coordinate transformations (orthogonal transformations) Now I know ...
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2answers
191 views

Understanding projective measurements as a special case of POVM measurements (“third postulate” in Nielsen and Chuang)

I am working through Nielsen and Chuang's book and am confused about a detail from sections 2.2.3 and 2.2.5. On page 88 of my copy (section 2.2.5), they write Projective measurements can be ...
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1answer
237 views

Problems finding the Eigenstates of a Hamiltonian

I am solving the problem 5.6 from Zettili Quantum Mechanics book: I have an Hamiltonian given by $\hat{H} = \epsilon \hat{\sigma} \cdot \hat{n}$ where $\hat{\sigma}$ represents the Pauli Matrices and ...
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2answers
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If $\sum_n \ c_n \ \psi_n(x,t)$ represents an arbitrary state for a given solution to the TISE, what are the bases for a free particle?

If $\sum_n c_n \psi_n(x,t)$ represents an arbitrary vector in the Hilbert space of solutions to Schrodinger's equation with a given potential function, this makes makes sense to me. Each $\psi_n$ can ...
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2answers
280 views

Orthogonal wave functions

I am wondering: can we explain the concept of orthogonality in physics for a beginner (without much math and linear algebra) by saying it simply means that the particle can not exist in two different ...
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1answer
167 views

Finding basis of Schmidt decomposition

How do you find the basis for the schmidt decomposition when given a state of multiple qubits? For example you have the systems A,B and C. Do they correspond to eigenvectors of a density matrix?
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How to handle bra-ket in logarithm?

$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{\bra}[1]{\left< #1 \right|}$ Say the following two equations: $$ S = - k_B \text{Tr} (\rho \ln \rho) $$ $$ \rho = \sum _\epsilon \ket{\...