2
votes
2answers
89 views
+50

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
0
votes
0answers
27 views

Proof that quantum relative entropy is $\leq$ 0 using Klein's inequality for positive semi-definite operators

I was asked to prove that $S(\rho) \leq - {\rm Tr} \left[ \rho \log \tau \right] $ where $\rho, \tau$ are density operators on a finite dimensional complex inner product space and $S(\rho)$ is the von ...
2
votes
0answers
49 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
0
votes
3answers
25 views

find a base to U Linear Algebra

dear users please help me... im answering a long question now ive been guided to find a base to U at the end of the process i got this $u= Sp\{x^4-3x^3+2x^2, 3x^4-7x^3+4x ,1\}$ and ive been guided to ...
0
votes
0answers
39 views

how to transform a quadratic equation into a matrix form?

I have this type of equation: $$ - a^ {2} A - \eta ^{2} B - a \eta C - b^{2} A' - \eta' ^{2} B' - b \eta' C' - a \eta' D - b\eta E $$ The capital letters, $A, A', B, B', C, C', D, E$ are just the ...
2
votes
1answer
82 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
1
vote
0answers
35 views

Linearize a specific eqution

Is it possible to linearize this equation ? I tried without success .. ...
1
vote
2answers
65 views

Trace in non-orthogonal basis

In Dirac notation we can define the trace of an operator in Hilbert space $\rho$ as the follows, $Tr(\rho)=\sum\limits_{|s\rangle \in B} \langle s| \rho |s\rangle$ where B is some orthonormal ...
3
votes
3answers
45 views

Planet Simulation Newtons Law - Downscaling

I would like to scale-down the original numbers of our planet's motion, as i cannot properly visualize it in Unity3D (Game-Engine). I have: 1) Initial Position (-3.5e10, 0) (km) 2) Initial ...
0
votes
1answer
54 views

Vector Space of Lie Algebra

Lie algebra $ \mathfrak{g} $ for a Lie group $ \mathcal{G}$ is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS). Firstly, when is it allowed to define an ...
0
votes
0answers
83 views

What is the reason for normalizing eigenvectors?

In Linear Algebra, when we have found eigenvectos related to specific eigenvalues, we normalize the eigenvectors. If I want to normalize eigenvectors, why do I need to normalize the eigenvectors?
0
votes
1answer
64 views

Finding Percentage Contribution of a Variable in an Equation

I have an equation, for example: $$ y=a-b+c $$ I am actually confused how exactly to find the contribution of the variables individually to the entire equation. Due to the negative sign, following ...
0
votes
1answer
60 views

Scaling a cup to have a certain filling volume

I created a cup in Autodesk Inventor using lathe/rotation, ie I defined the profile and rotated it around an axis. I measured it's volume. By using Patch and Sculpt I filled the inner volume(which ...
4
votes
1answer
122 views

Linear Algebra in curved space

We know that Euclidean geometry and Newtonian Physics are special cases that only work in a flat space-time. Got to thinking about linear algebra and matrices. Is linear-algebra a special subset of ...
1
vote
0answers
67 views

Transpose of an operator $T$

How can I prove that a transpose operator is a basis-dependent? Is it true that I define transpose operator in this way: $ A^T= \Sigma_{i,j} \langle e_j|A|e_i\rangle$?
1
vote
1answer
48 views

Finding the kernel , the image and the rank of $[A\ A]$ for an invertible $A$

Let $A$ be an invertible matrix of order $n$. What are the kernel, the image, and the rank of the matrix $\begin{bmatrix} A & A \end{bmatrix}$ (of order $n \times 2n$)?
2
votes
0answers
39 views

Fock Subspaces and Weight Vectors

I've got an assignment due in a few hours, and I'm at a complete loss as to how to even start it, really. I haven't encountered any Dirac notation before, so I'm having a lot of trouble attempting the ...
0
votes
0answers
41 views

Equations of motion for a block

I am looking for a very simplified derivation of the equations of motion (rotational and translational) for a block with a body fixed frame. I need to compare the EOMs for a system when the center ...
0
votes
2answers
20 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
1
vote
0answers
65 views

Geometric or physical meaning of a defective matrix

I've been reading wikipedia pag of Jordan canonical form, which induces matrices that does not have eigenbasis, i.e. defective matrices. The physical and geometric meaning of normal matrices are ...
2
votes
1answer
76 views

Affine space $A^n$ and definition of difference.

I'm not sure if this question would be more appropriate in Physics.SE, if so let me know. I need help in understanding this quote from "Arnold - Mathematical Methods in Classical Mechanics" (This is ...
0
votes
1answer
60 views

Write the normal and vector form of the equation in $\mathbb{R}^2$

This is more of a check then anything else. Here is what I have. Need to find the normal and vector form of the equation $$-2x+3y=5$$ Normal form: $$(-2,3) \cdot [(x,y) - (-1,1)]$$ Vector form: Now ...
3
votes
1answer
41 views

For nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, is the length of the projection of $v$ along $u$ always less than the length of $v$.

For all nonzero vectors $u,v\in\mathbb{R}^2$, where $u\neq v$, the length of the projection of $v$ along $u$ is less than the length of $v$. This is a true/false question and when I said true it ...
2
votes
3answers
75 views

The matrix notation of signum?

The following question on a notation might look trivial but I am really not sure how to deal with it. If I have a variable $x$, I could write out: $$x=|x|\;\text {sgn} (x)$$ a notation that helps ...
3
votes
0answers
73 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
0
votes
0answers
146 views

What does the partial trace of an operator tell me about the full operator?

I have a situation where I would like to know something about an operator when I know something about its partial trace. Let $A$ be a trace class operator on $H\otimes H$, nonnegative, of trace one, ...
7
votes
1answer
343 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
2
votes
0answers
89 views

Solving Generalized Eigenvalue Problem perturbatively

Let me formulate the problem to convey my notation. I have a matrix $A$ which is hermitian - and is diagonalisable by a transformation $$ U_A A\,\,U_A^{-1} = A_{diag}$$ Now the matrix is changed, ...
1
vote
1answer
85 views

The representation of the resolvent of a quadratic form

I know some aspects are related each other concerning resolvent ,such a system of linear equation with a parameter, Fredholm theory and Green function method in nonlinear equation when I am reading ...
12
votes
3answers
2k views

What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
0
votes
1answer
153 views

Find the parametric equation of the path of an object going at a speed x , with an orientation vector at time t and point p

$$ x = 1/2, v[2,−2], t = 3, P(1, 2)$$ x = speed, v = orientation, t = time, p is a point. I tried this : $$((2, -2) - (1,2)) \sqrt{1^2 + 4^2} $$ I got my <1,4> from $(2-1, 2-(-2))$ , which is my ...
0
votes
1answer
42 views

What is $\left(\delta_{ab}\right)^{-1}$?

I have an expression that involves the Wigner 3j coefficient: $$\left(\matrix{a&b&0\\0&0&0}\right)^{-1}$$ This simplifies to: ...
2
votes
1answer
154 views

Invariant polynomials

Let $A\in Mat_{2}(2\times2;\mathbb{R})$and consider the action of the 1-parameter group $e^{tA}$ on $\mathbb{R}^{2}$. Describe all 1-parameter groups etA which have a nonconstant invariant polynomial. ...