Matrix Definition and 1000 Threads

Hello! I really don't know much about statistics, so I am sorry if this questions is stupid or obvious. I have this data: ##x = [0,1,2,3]##, y = ##[25.885,26.139,27.404,30.230]##, ##y_{err}=[1.851,0.979,2.049,6.729]##. I need to fit to this data the following function: $$y = a (x+0.5)/4.186 +...

The below matrix represents a rotation. 0 0 -1 0 1 0 1 0 0 Im trying to obtain the general point (x y z) when rotated by the above rotation matrix? So visited https://www.andre-gaschler.com/rotationconverter/ entered the above figures and not sure which entry would be x y z but assume it...

I think I roughly see what's happening here. > First, I will assume that AB - BA = C, without the complex number. >Matrix AB equals the transpose of BA. (AB = (BA)t) >Because AB = (BA)t, or because of the cyclic property of matrix multiplication, the diagonals of AB equals the diagonals of...

Hey! :o We have the matrix $A=\frac{1}{3}\begin{pmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{pmatrix}$. Show that there is an unit vector $v_1$, such that $A=I-2v_1v_1^T$. We consider an orthogonal matrix $Q=\begin{pmatrix}v_1 & v_2 & v_3\end{pmatrix}$. Show that...

Suppose I have a model composed of two parameters ##(a,b)## that I want to describe a set of data points with. In CASE A, I fit the model taking into consideration the correlations between the data points (that is, in the chi square formulation I use the full covariance matrix for the data) and...

Below is the attempted solution of a tutor. However, I do question his solution method. Therefore, I would sincerely appreciate it if anyone could tell me what is going on with the below solution. First off, the rotation of the matrix could be expressed as below: $$G = \begin{pmatrix} AB & -||A...

I'm fairly stuck, I can't figure out how to start. I called the matrix ##\mathbf{A}## so then it gives us that ##\mathbf{A}\mathbf{A}^\intercal = \mathbf{I}## from the orthogonal bit. I tried 'determining' both sides... $$(\det(\mathbf{A}))^{2} = 1 \implies \det{\mathbf{A}} = \pm 1$$I don't...

Hey! :o We have the vectors $\displaystyle{a_k=\begin{pmatrix}\cos \frac{k\pi}{3} \\ \sin \frac{k\pi}{3}\end{pmatrix}, \ k=0, 1, \ldots , 6}$. Let $P_k$ be the projection matrix onto $a_k$. Calculate $P_6P_5P_4P_3P_2P_1a_0$. Are the elements of the projection matrix defined as...

I want to solve the following system of equations ##M_{1} = f_1+f_2+m_1+m_2\ \ ;\ \ M_{7} = f_1+f_2+s_1+s_2\ \ ;\ \ M_{13} = m_1+m_2+s_1+s_2## ##M_{2} = f_1+f_3+m_1+m_3\ \ ;\ \ M_{8} = f_1+f_3+s_1+s_3\ \ ;\ \ M_{14} = m_1+m_3+s_1+s_3## ##M_{3} = f_1+f_4+m_1+m_4\ \ ;\ \ M_{9} = f_1+f_4+s_1+s_4\...

$$\hat{S_+} = \hbar \begin{bmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{bmatrix}$$ $$\hat{S_-} = \hbar \begin{bmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{bmatrix}$$ $$\hat{S_x} = \hbar/\sqrt{2} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0...

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...

Hello! (Wave) The general solution of the system $Ax=\begin{bmatrix} 1\\ 3 \end{bmatrix}$ is $x=\begin{bmatrix} 1\\ 0 \end{bmatrix}+ \lambda \begin{bmatrix} 0\\ 1 \end{bmatrix}$. I want to find the matrix $A$. I have done the following so far: $$x=\begin{bmatrix} 1\\ 0 \end{bmatrix}+...

I would like to solve a system of systems of equations Ax=b where A is an n x m x p tensor (3D) matrix, x is a vector (n x 1), and b is a matrix (n x p). I haven't been able to find a clear walk-through of inverting a tensor like how one would invert a regular matrix to solve a system of linear...

Evening, The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest. Here are the [questions](https://prnt.sc/riq7m0) and here are the...

Hello, I'm currently writing a numerical simulation code for solving 2D steatdy-state heat conduction problems (diffusion equation). After reading and following these two book references (Numerical Heat Transfer and Fluid Flow from Patankar and And Introduction to Computational Fluid Dynamics...

I am using the following code. It's returning the block matrix (Z) raised to negative one (think about inputting 22/7 in a Casio fx-991ES PLUS). import sympy as sp from IPython.display import display X = sp.Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3]]) i = sp.Matrix([[1], [1], [1]]) Z =...

Hi all. A problem has arisen whereby I need to maximize a function which looks like $$ f(A) = \mathbf{w}^T \left[\int_0^t e^{\tau A} M e^{\tau A^T} d\tau \right]^{-1} \mathbf{w} $$ with respect to the nxn matrix A (here, M is a covariance matrix, so nxn symmetrix and positive-definite, w is an...

One way to express a function of a matrix A is by a power series (a Taylor expansion). It is not too difficult to show that two functions f(A) and g(A) with such a power series representation must commute, i.e. f(A)g(A) = g(A)f(A). But matrices typically do not commute with their own transpose...

As part of a group project, we have been asked to create a code which produces the following matrix: I want to create the graphics of this in a way that looks similar to this: But of course with different number of cells, different colours, etc. The problem is that from what I have found...

Hello! I have checked commutator matrix form of $$\vec{p}=im/\hbar [H,\vec{x}]$$ but I realized i don't undertand something I have $$[H,\vec{x}]=H\vec{x}-\vec{x}H$$ and $$(H\vec{x})_{i}=H_{ij}x_j$$ & $$\ ( \vec{x}H)_{i}=x_jH_{ji}$$ but what is the second term matrix representation...

Hello! I am new here, and I need (urgent) help regarding the following question: Let $\boldsymbol{A}_{(n\times n)}=[a_{ij}]$ be a square matrix such that the sum of each row is 1 and $a_{ij}\ge0$$(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$ are unknown. Suppose that...

I am not sure if this is the right forum to post this question. The title says it all: are there examples of Lie groups that cannot be represented as matrix groups? Thanks in advance.

(scroll to bottom for problem statement) Hello, I am wondering if someone could break down the problem statement in simpler terms (not so math-y). I am struggling with understanding what is being asked. I will try to break it down to the best of my ability Problem statement:Consider the inner...

Hey! :o We have the matrices \begin{equation*}a=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}, \ \ E_{1,3}=\begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \ \ u_n=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & 1\end{pmatrix}\end{equation*} I have...

Suppose the Bell operator ##B=|AB(1,2)+AB(1,3)+AB(2,3)|## With ##AB\in{1,-1}## Nonlocal realism implies ##B\in{1,3}## However using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1, implying nonlocal realism cannot explain the quantum result. However if...

Hi! I don't understand how to demonstrate the following exercise. Let ##F: R^{n} \rightarrow R^{n}## be a linear map which is invertible. Show that if ##A## is the matrix associated with ##F##, then ##A^{-1}## is the matrix associated with the inverse of ##F##.

I have been teaching myself matrices for my IGCSE course and I ran into a problem in a past paper which I have no clue how to solve. The problem is part b of the attached image. Thanks for your help in advance.

Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this: \begin{pmatrix}...

I am familiar with the hessian matrix having the square in the numerator and a product of partial derivatives in the denominator: $Hessian = \frac{\partial^2 f}{\partial x_i \partial x_j}$ However, I have come across a different expression, source...

Hi, In the question outlined in the images (apologies for the poor quality of the scans), the chosen solution has opted to use a symmetry argument and proceed from there. Question is from "Structures: theory and analysis" by Williams & Todd My question is: How could we approach the same...

The Fourier transform of a function is called its spectrum. The set of eigenvalues of a matrix is also called a spectrum. Why the same name? Is there some hidden connection between these two?

I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...

My idea was to write out the formulas for the orthogonal q vectors in terms of the input vectors using the basics of gram-schmidt. Then, I would rewrite those equations suhc that the a vectors were written in terms of the q vectors. And then, try to find some matrix which would capture the...

Hello, everyone. :) All I could gather is that, if I'm correct, lattices are spans of the column vectors of the matrix within the "LAT()" notation and the X and Y occurrences are unit placeholders (such as the pixel unit (since this is in the context of image processing)). And, as an attempt...

vector=(abc) 1. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0& sin(\theta) & cos(\theta) \end{pmatrix}$$ The rotation part is correct. 2. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& 0 & 0 \end{pmatrix}$$ is wrong apparently how do I do the mirroring? step 3 i can do...

Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening: $$ \left( \begin{array}{cccc} \frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\ 0 & 1 & 0 & 0...

This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...

hello i hope everyone doing well, I have problem in Stiffness Matrix For Beam element (2 nodes ) i have a beam element i want to get a stiffness matrix: we have beam element (2 nodes) node (1) : u1 horizontal displacement, v1 vertical displacement node (2): u2 horizontal displacement , v2...

In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...

Help me if what I have done so far can be simplified further.

Summary: different methods give different results. They are not consistent. Summary: different methods give different results. They are not consistent. I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...

I have unfortunately no attempt for 31.i. I don't know how to start the problem. 31.ii is attached. Can someone give me a tip on how to start 31.i? I also have troubles solving 29.iii. I have attached that as well.

I am conducting research into the matrix exponential, and I would like to discuss the mathematical development of the exponential, and eventually some applications in terms of quantum physics. Currently, my problem is tracking down the original usage of this technique. I am trying to find a...

If I am given that the density matrix of an incoming beam of spin 1 particles of the form $$p_o=(1/3)[|1><1|+|0><0|+|-1><-1|]$$, aand I needed to find the fraction of particles that would be found with a spin x component of zero, how would I go about solving this problem? My hunch is that I...

Hello! (Wave) Let a matrix with $n$ integers in the interval $[1,100]$. I want to write an algorithm that sorts the elements of the matrix in $O(n)$. [Hint: Count and use the number of times that each number appears] I have thought the following: We create a matrix with $100$ entries. If the...

The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...

I've already posted this question on the mathematics website of stack exchange, but I've received more help here in the past so will share it here as well. I am developing a tool for missile trajectory (currently without guidance). One issue is that the aerodynamic equations on the missile are...

I know that every ##m×n## matrix ##D## can be expressed in the form $$P_1DP_2 = LU$$ where ##P_1## and ##P_2## are permutation matrices, ##L## is a unit lower triangular matrix, and ##U## has the form $$\begin{bmatrix} U_1 & U_2 \newline 0 & 0 \newline \end{bmatrix}$$ where ##U_1## is an...

I feel confused about proving the two terms are idempotents.