Matrix Definition and 1000 Threads

Hi, there. I am working with a model, in which the dimension of the Hilbert space is infinite. But Since only several states are directly coupled to the initial state and the coupling strength are weak, then I only consider a subspace spanned by these states. The calculation shows that the...

In the 4-dimensional representation of ##\beta##, ## \beta=\begin{pmatrix}\mathbf I & \mathbf 0 \\ \mathbf0 & -\mathbf I\end{pmatrix} ,## and we can suppose ## \alpha_i=\begin{pmatrix}\mathbf A_i & \mathbf B_i \\ \mathbf C_i & \mathbf D_i\end{pmatrix} ##. From the anti-commutation relation...

https://projecteuler.net/problem=101import numpy as np for j in range (1,11): M = np.empty([j, j]) for x in range(1,j+1): for y in range(1,j+1): M[y,x] = y**(j-x) Minv = np.linalg.inv(M)The ##j^{\mathrm{th}}## estimate ##\mathrm{OP}(j,n)## which fits ##j## data...

The following matrix is given. Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1. The answer sheet reads that the diagonal matrix D is as follows: I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...

Hey! A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$. Show that a twice continuously differentiable funtion $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex iff the...

Hello, I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A). Then, how can we prove that...

My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution). As the first column is all 0s, x1 would be a free variable. Because the system with free variable have infinite solution, the solution is not unique. In this way, the matrix is...

Hi, I'm using Scientific Workplace to write LaTex and it generates the code shown below for the given matrix. I don't think the generated code is standard LaTex in this particular instance. How can I fix it without making too many modifications? I mean if there is a simple way to fix it. Thank...

hello matrix and wave formulation of QM are equivalent theories i.e they yield the same results Which one is most frequentely used by professional scientists in solving real problems and why ?

I have taken the variables as follows: A[][]=the matrix max=to store the maximum integer value present in the matrix min=to store the minimum integer value present in the matrix sum=to store the sum of boundary elements display()=methos to print matrix sort()=method to sort matrix in descending...

Good afternoon to all again! I'm solving last year's problems and can't cope with this problem:( help me to understand the problem and find a solution!

good evening everyone! Decided to solve the problems from last year's exams. I came across this example. Honestly, I didn't understand it. Who can help a young student? :) Find characteristic equation of the matrix A in the form of the polynomial of degree of 3 (you do not need to find...

Gonna preface by saying I never thought linear algebra would be a class I would regret not taking so much... but in short the goal is to reduce an arbitrary symmetric NxN system using a set of auxiliary constraint relationships, e.g. for a 3x3 \begin{bmatrix} V_1\\ V_2\\ V_3\\ \end{bmatrix} =...

In a 2012 article published in the Mathematical Gazette, in the game of golf hole score probability distributions were derived for a par three, four and five based on Hardy's ideas of how an hole score comes about. Hardy (1945) assumed that there are three types of strokes: a good (##G##)...

Hello, Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There...

I try to solve but i have 1 step in the solution that I don't understand who to solve. Below in the attach files you can see my solution, the step that I didn't make to prove Marked with a question mark. thanks for your helps (:

Hi, I was trying to do the following problem. I was able to do the part in pink highlight (please check "My attempt") but the part in orange highlight makes no sense to me. I'd really appreciate if you could help me to solve the part in orange. Thank you! My attempt: The solution presented...

$$\langle p | W | p' \rangle = \int \langle p | x \rangle \langle x W | x' \rangle \langle x' p' \rangle dx dx'$$ $$\langle p | W | p' \rangle = \int \langle p | x \rangle \delta(x-x') W(x) \langle x' | p' \rangle dx dx'$$ $$\langle p | W | p' \rangle = \int \langle p | x' \rangle W(x') \langle...

Hi, It's actually not a homework problem but I still decided to post it here. Problem: Consider Ax=y, where A is mxn and has rank m. Is (A′A)⁻¹A′y a solution? If not, under what condition will it be a solution? Is A′(AA′)⁻¹y a solution? The given solution is: Consider Ax=y with A mxn and...

Mentor note: Since the technique used here involves differentiation, I moved this to the Calculus section. Hi, I was trying to do the problem below. I was following the approach presented in this answer. I assume the approach is correct. The answer I ended up with is clearly wrong. Where am I...

I found three projection operators $$P_{1}= \begin{pmatrix} 1/2 & & \\ & -\sqrt{2}/2 & \\ & & 1/2 \end{pmatrix}$$ $$P_{2}= \begin{pmatrix} 1/2 & & \\ & \sqrt{2}/2 & \\ & & 1/2 \end{pmatrix}$$ $$P_{3}= \begin{pmatrix} -1/\sqrt{2} & & \\ & & \\ & & 1/\sqrt{2} \end{pmatrix}$$ From this five...

Hello! I need to calculate the ABCD matrix for a thick concave mirror, in the situation in which the light comes from the plane side of the mirror, and it is the concave part that is coated (for reference, I have a Fabry Perot cavity with 2 concave mirrors, and I want to mode match the laser...

Hi, I was trying to find the rank of following matrix. I formed the following system and it seems like all three columns are linearly independent and hence the rank is 3. But the answer says the rank is '2'. Where am I going wrong? Thanks, in advance!

i know that there is the Cayley -Hamilton theorem but i don't know if i can use it and how.Do you have any ideas about it?Please give me any help.

https://drive.google.com/file/d/1g7fjWAUEpOo2NukqFqZI4Wrujud6sjbn/view?usp=sharing $\tiny{4.288.T20}$ Suppose that A is a square matrix of size n and $\alpha \in \CC$ is $\alpha$ scalar. Prove that $\det{\alpha A} = \alpha^n\det{A}$. Using $\alpha=5$ $\det{5A}=\det\left(5\left[...

Hi, I have a 3 mass system. ##M \neq m## I found the forces and I get the following matrix. I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values. I'm wondering if...

Hello, I am looking for a worked out solution to diagonalize a general 4x4 Hermitian matrix. Is there any book or course where the calculation is performed? If not, does this exist for the particular case of a traceless matrix? Thank you!

I have successfully found the N by N matrix corresponding to the operator R. But the problem is, whenever I try to operate R on |bj> basis vectors, I am not getting |b(j+1)> as it should be. Instead, I am getting result as given in the question only by <bj|R = <b(j+1)| Matrix is not working...

Hi everyone I am new to this or any online math board community, I’m looking for assistance in determining and calculating the size of a lottery draw pattern matrix, using simple mathematics formulas. That I myself can learn to use to include math formulas on the information pages of my new...

Hey! :giggle: Let $\lambda\in \mathbb{R}$ and \begin{equation*}a=\begin{pmatrix}1 & 2 &-1& \lambda & -\lambda \\ 0 & 1 & -1& \lambda & 2\\ 2 & 2 & 1 & 1 & 3\lambda-1 \\ 1 & 1 & 1 & \lambda & 5\end{pmatrix}\in \mathbb{R}^{4\times 5}\end{equation*} (a) Let $\lambda=1$. Determine a Basis...

Trying to use <+|+>=1=<-|-> and <-|+>=0 to prove each iteration of the equation, so I have 6 different versions to prove. But the part I'm currently stuck on is understanding how to simplify any given version. I've written out [S_x,S_y]=S_xS_y\psi-S_yS_x\psi and expanded it in terms of the...

Hey! :giggle: We have the following linear maps \begin{align*}\phi_1:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto \begin{pmatrix}x+y\\ x-y\end{pmatrix} \\ \phi_2:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto...

Hi, suppose I am given an SL(2C) matrix of the form ##\exp(i\alpha/2 \vec{t}\cdot\vec{\sigma})## where ##\alpha## is the complex rotation angle, ##\vec{t}## the complex rotation axis and ##\vec{\sigma}## the vector of the three Pauli matrices. I would like to decompose this vector into...

While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for xT A x, where x...

Also, if it's possible, I would really like to know the command for inputting this kind/type of problem on Ti-89 in order to check correct answers for linear algebra problems like this one.

Find determinant of following matrix: ## A = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n-1} & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n-1} & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n,1} & 0 & \cdots & 0 & 0 \end{pmatrix} ## Note: I tried to solve this question...

I'm working on the time-dependent Schrodinger equation, and come across something I don't understand regarding notation, which is not specific to TDSE but the Schrodinger formalism in general. Let's say we have a non-trivial potential. There is a stage in the development of the TDSE where we...

Matrix multiplication is defined by \sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw \sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij} This is because ##A^TA=I##. How to know how many independent parameters we have in...

I think you all can see that ##a_{(i+1,j+1)} = a_{i,j} + a_{i+1,j} + a_{i,j+1}## Now the determinant always give me problem. I have and idea to reduce this matrix by Chio to a 2x2 matrix and find the determinant of this 2x2. Put i was not able to see any pattern to find what how the 2x2 matrix...

Hello, I am studying change of basis in linear algebra and I have trouble figuring what my result should look like. From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...

I'm reading the book QFT by Ryder, in the section where ##\rm{SU(2)}## is discussed. First, he considered the group of ##2 \times 2## unitary matrices ##U## with unit determinant such that it has the form, $$U =\begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix}, \qquad \xi = \begin{bmatrix}...

Summary:: I am suppose to show that this columns matrix does not transform as a vector. In another words, it is not in fact a vector. I think this become trivial if we get the rotation matrix composed of Euler angles. But, i think that it is not the best way to solve this problem, and i...

$\tiny{311.2.2.6}$ Use the inverse to solve the system $\begin{array}{rrrrr} 7x_1&+3x_2&=-9\\ -2x_1&+x_2&=10 \end{array}$ the thing I could not get here without a calculator is $A^{-1}$

What's the correct command for finding an LU factorization of a 3x3 and 4x4 matrix on Ti-89 graphing calculator? I'm trying to find the correct answers and verify/check my answers for Linear Algebra problems.

With ##\rho=\sum_i p_i|\Psi_i\rangle\langle\Psi_i|##If the ##p_i=|\langle\Psi|\lambda_i\rangle|^2## are taken as joint probabilities given by quantum mechanics for the singlet state in EPRB then this cannot represent a statistical mix (classical) of those states because of Bell's theorem ?

But I actually don't get the same matrix. What I get is the transpose of the other when I change the order i.e when I do [A]^2[A] I get the transpose of [A][A]^2 and vice versa What I'm trying to do is find the cube of the expectation value of x in the harmonic oscillator in matrix form. We're...

Hey! 😊 Calculate the Cholesky decomposition of the matrix, the only non-vanishing elements are the diagonals $1,2,3, \lambda$ and all under and upper secondary diagonal elements are $1$. For which $\lambda$ is the matrix singular? Could you please explain the form of the Matrix? Does the...

I have solved the exercise, so I'm not giving the vectors explicitly. I just want to know if there is a quicker way than mine. We know that ##A## must have ##4## columns and ##4## lines, and we also know that its nullity is ##2##, thus its rank is ##2##. I took the simplest matrix that can have...