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

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1answer
69 views

Special relativity: how to prove that $g = L^t g L$?

We have $$X^\textrm{t}gX = 0 \iff X^\textrm{t}L^\textrm{t}gLX = 0,$$ where $X$ is a column vector of length four, $L$ is a non-singular $4 \times 4$ matrix, 't' denotes matrix transpose, and $$g = ...
3
votes
2answers
355 views

Eigenenergies and eigenkets given the Hamiltonian

For a two level system the Hamiltonian is: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\rangle\langle2|+|2\rangle\langle1|) $$ where $a$ is a number with the dimension of an energy. I need to ...
3
votes
1answer
66 views

Boson calculus and Maximum Weight State

I'm just going over a few past exams for tomorrow, and I've come across a question that I'm having quite a bit of difficulty with. Let $\left|0\right\rangle$ denote the Fock vacuum state so that ...
3
votes
1answer
212 views

Random Hankel matrix and eigenvalues distribution

I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices.
3
votes
1answer
81 views

Complex semi-definite programming

I'm doing some calculations and I want to simulate them in python or matlab (or whatever). However I use hermitian matrices and I don't really manage to find a library which enables me to calculate ...
3
votes
1answer
77 views

Mutually unbiased bases

This question can be formulated in two ways. Let there be two $d$-dimensional orthonormal bases $B_{1}$ and $B_{2}$. I refer to the elements of $B_{1}$ by $\lvert\nu_{i}\rangle$ and to the elements of ...
3
votes
1answer
509 views

Rotation about the $z$-axis on the Bloch sphere

I'm having some trouble with successfully working out a rotation about the $z$-axis on the Bloch sphere. Now, I know how this is performed, in principle. A rotation of the Bloch-sphere around an ...
3
votes
1answer
577 views

1D Ising Model (NN and NNN interactions) with 2 transfer matrices

I've tried an alternative solution for finding the partition function of this model. So is what I've done correct? If it isn't then please prove and explain why not. (I'm pretty sure I made a ...
3
votes
1answer
116 views

Are the symmetry operators well defined in the context of Projective Symmetry Group(PSG)?

Consider the Schwinger-fermion approach $\mathbf{S}_i=\frac{1}{2}f_i^\dagger\mathbf{\sigma}f_i$ to spin-$\frac{1}{2}$ system on 2D lattices. Just as Prof.Wen said in his seminal paper on PSG, the ...
3
votes
1answer
246 views

Algorithm for identifying planes in a Bravais Lattice

I have a lattice with Lattice Vectors $(\vec{t}_1,\vec{t}_2,\vec{t}_3)$ which are NOT orthogonal in general. How can I identify the atoms/unit cells that belong to a plane - that is normal to a given ...
3
votes
2answers
281 views

How to express continuous values as a matrix

Usually a quantity of a matrix is defined as the eigenvalues of the matrix. If so, how can anyone express continuous values, as in Schrodinger picture, into a matrix?
3
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0answers
65 views

Geometry topics in physics [closed]

I'd like to learn modern physics at an advanced level, but since I've no access to university, I'm self-teaching, and appeal to the Internet for information about what to study and how. Currently, ...
3
votes
0answers
66 views

How coordinate system shifting is related to similarity transformations?

I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?
3
votes
1answer
165 views

Determinant and adjunct of $k-\omega^2m$ in terms of natural frequencies

Given is a mechanical multiple degree of freedom system described by the following matrices and equation: mass matrix ${\bf{m}} = \left[\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 ...
3
votes
0answers
155 views

Fock Subspaces and Weight Vectors

This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons. I've got ...
2
votes
3answers
229 views

Dimension of vector resulting from tensorial product

I'm quoting what I found in a book about quantum computation: Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then ...
2
votes
3answers
311 views

How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
2
votes
3answers
111 views

Diagonalization Of $(\sigma_x+\sigma_y)$

Can this matrix $(\sigma_x\pm\sigma_y)$ be diagonalised? Clearly, if $\sigma_x$ is diagonalized by a similarity transformation $S_1\sigma_x{S_1}^{-1}$, then $\sigma_y$ can't be diagonalized by $S_1$, ...
2
votes
3answers
360 views

Vector space of $\mathbb{C}^4$ and its basis, the Pauli matrices

How do I write an arbitrary $2\times 2$ matrix as a linear combination of the three Pauli Matrices and the $2\times 2$ unit matrix? Any example for the same might help ?
2
votes
1answer
186 views

Infinite-dimensional Hilbert spaces in physical systems

Can anyone give an example of when infinite-dimensional Hilbert spaces are required to describe a physical system? The standard answer to this question is yes, and I'm sure some of you will be quick ...
2
votes
2answers
151 views

When is matrix representation of a Hermitian operator invertible?

If I have a Hermitian operator $H:V \to V$ on a finite-dimensional vector space $V$, and I write down its matrix representation in some basis $B$ with matrix representation being $[H]_B$, then in what ...
2
votes
1answer
91 views

Extension of Lami's theorem

I was experimenting with the triple scalar product and forces in equilibrium when I came to this result: Consider 4 forces $ \pmb{F_i}$ for $i=1,2,3,4$. $\pmb{F_i}=F_i\hat{e_i}$ where $\hat{e_i}$ is ...
2
votes
2answers
364 views

Differences between wave function and set of orthonormal wave functions?

I'm reading a QM book. It first says for wave function: "The state of a physical system (or particle) is completely specified by an entity associated with it called a wave function, Ψ , that in ...
2
votes
1answer
52 views

Local fermionic symmetry and GS action

I have a trouble understanding an argument which I think has a simple answer but I am not getting it. The question is that if you don't impose local fermionic symmetry the GS action has only one term ...
2
votes
1answer
94 views

Characteristic polynomial of a Matrix

In fact, this problem is more likely to be a math problem. When I read a paper(http://arxiv.org/abs/0707.2875), the author includes the characteristic polynomial for a type of matrix $A_k$ with ...
2
votes
2answers
70 views

How to take partial trace?

$L$ is a linear operator acting on hilbert space $V$ of dimension $n$, $L: V \to V$. The trace of a linear operator is defined as sum of diagonal entries of any matrix representation in same input ...
2
votes
1answer
60 views

Can All 4-D column matrices be given as tensor product of 2-D column matrices?

I am familiar with entanglement concept. But it feels bit weird to me that all possibilities of a system in a $4$-dimensional vector space cannot be given as tensor product of two $2$-dimensional ...
2
votes
2answers
107 views

Tensors as multilinear maps

Sean Carrol's in his book on GR introduces tensors as a multilinear map of a set of dual vectors and vectors onto R. I usually think of tensors as a multidimensional array of numbers with fixed ...
2
votes
2answers
107 views

Expanding a ket in the position basis?

My textbook says that to find the ket $|ψ\rangle$ in the same position basis as the ket $|ø\rangle$ we do the following: $$|ψ\rangle=\int dø|ø\rangle \langle ø|ψ\rangle$$ Firstly can $|ø\rangle$ be ...
2
votes
1answer
61 views

Formular for interior product, example

In Nakahara's Geometry,Topology and Physics, the interior product is defined like this : $$i_X: \Omega^{r}(M) \rightarrow \Omega^{r-1}(M).$$ Where $ X \in X(M)$ and $\omega \in \Omega^{r}(M)$ ...
2
votes
3answers
144 views

Shankar's Active/Passive Change of Basis

I'm working my way through Shankar's Quantum Mechanics (7th printing, and I'm doing it alone, so I apologize if I have core concepts completely wrong). He has a section on Active and Passive ...
2
votes
1answer
269 views

What is the physical interpretation of the dot/inner/scalar product of two vectors?

What is the physical interpretation of the dot/inner/scalar product of two vectors? See, if we multiply two scalars like 2*3 we say two times three is six. I also do understand multiplication of ...
2
votes
1answer
99 views

Positive cone of operators: if two selfadjoints $a$, $b$ obey $a^2 + b^2 =1$, must they commute?

The question relates to the structure of the positive cone of operators, in C*-algebra. If $a$ and $b$ are selfadjoints such that $a^2 + b^2 = 1$ can one prove $a$ and $b$ commute? What one ...
2
votes
3answers
2k views

Spring-mass system with two springs and three masses [closed]

I'm trying to solve a system of springs and masses that is confusing me. First, the balls are all lined up linearly. Secondly, the ball in the middle has a smaller mass $m$ while the first and last ...
2
votes
1answer
90 views

Why eigenvector points to principal stress plane?

I can represent a tensor by a matrix. Suppose we are talking about a 2nd order tensor, and the matrix is therefore 3x3. If I find one eigenvector of that matrix; that vector represents normal vector ...
2
votes
1answer
94 views

Determinant expansion

I've seen in a few textbooks now a useful looking expansion procedure for determinants, but I don't understand the details of it. Here is precisely the example I'm thinking of (Ex. 7.6). I don't ...
2
votes
1answer
76 views

Schriffer Wolff Transformation - for first order change in eigenvalues

Step 1 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 ...
2
votes
3answers
934 views

What is the general approach to calculating time of impact in 3D?

Given two objects a and b moving at fixed velocities, how would you determine (a) whether they will collide at all, and if so, (b) time of impact? (Let us assume these are spherical bodies each with ...
2
votes
0answers
38 views

commutation relations for operators in projected subspaces

I am looking for a consistent re-definition of commutators for certain operators when I work in a projected subspace. Basically, I have a spin defined in terms of 4 Majorana operators $b_{x}$, ...
2
votes
2answers
103 views

What's the degree of freedom of this kind of matrix?

We first have a unitary matrix $$\{a_{ij}\}\quad(n\times n)$$ I know how to calculate its degree of freedom, which is $n^2$ if we consider a real variable as one degree of freedom. Now we have a ...
2
votes
0answers
16 views

Dropping vertices of an overdetermined statics system graph

In statics, the problem of determining the tensions of K cables that connect a structure made of N points and keep all points in static equilibrium implies $ND$ systems of equations, where $D$ is the ...
2
votes
0answers
252 views

What is the physical meaning of complex eigenvalues?

I understand the mathematical origin of complex eigenvalues, and that complex eigenvalues come in pairs. But what is the meaning of the imaginary part? In particular I refer to an acoustic problem ...
2
votes
0answers
77 views

Could the phase factor $i$ be replaced by “matrix representation” totally in quantum mechanics? [duplicate]

It seems that $i$ plays an important role in quantum mechanics (Q.M.). On the other hand, linear algebra plays such an important role in Q.M. too. So would linear algebra, such as a matrix be able to ...
2
votes
0answers
44 views

Linear Algebra For Physicists (Book Recommendations) [duplicate]

I am aware that there are plenty of questions regarding book recommendations, however, I have not found one that fully matches what I intend to ask. I have provided a list of links to some similar ...
2
votes
1answer
135 views

Eigenvalues of a mean correlation matrix (integral over correlation matrices with arbitrary density)

Consider a stationary dynamic system with state $s(t)$ and correlation structure described by $C_{ij}(\tau)=\mathbb{E}[(s_i(t+\tau)-\bar{s_i})(s_j(t)-\bar{s_j})]$. Given an arbitrary density function ...
2
votes
2answers
614 views

2D - Kinematics - Linkage System using Vector Algebra [closed]

I have this question that I dont know how to solve correctly : My question is, how do I find $V_B$ ? I will find the angular velocities myself, but I want to know the method to get $V_B$ ? I know ...
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vote
5answers
612 views

Physical applications of matrices and determinants

Other than notation devices, I don't see any direct application of matrices/determinants in physics. For example, they are just a different way to write a partial derivative and determinants find if ...
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vote
4answers
684 views

Vectors with more than 3 components

I have some confusion over Vectors, Its components and dimensions. Does the number of vector components mean that a vector is in that many dimensions? For e.g. $A$ vector with 4 components has 4 ...
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vote
2answers
109 views

$\exp(i\alpha\hat {\bf n}\cdot{\bf \sigma} )=\cos\alpha I+i(\hat {\bf n}\cdot{\bf \sigma})\sin\alpha$

Could anyone tell me $\hat {\bf n}\cdot{\bf \sigma}$ is defined in such way? In the book they have not defined what is $n_z,n_x,n_y$. It is from Quantum Computing: From Linear Algebra to Physical ...
1
vote
1answer
343 views

Why consider only direction cosines?

Why are these called direction angles? Why do we consider only direction cosines and not direction sines or tans. What is its actual significance? And How to use them? Why are they called ...