Matrix Definition and 1000 Threads

The theorem reads as follows: Proof. The proof is by induction on the number of columns of an ##m\times n## matrix ##A##. The base case is kind of clear (either the rref is the ##0## vector or ##e_1##). Suppose now that the rref of a matrix with ##k## columns is unique and consider ##A## with...

A matrix nxn with a parameter p is given and the question is what is the rank of that matrix in terms of p, the gaussian elimination is the standard process and i know how to do it. But i was wondering if the determinant of a matrix tells us if the matrix has independent columns thus telling us...

Hi, the task is as follows I had no problems deriving the expressions ##\omega##, ##\frac{\textbf{S}}{c}## and ##\frac{\textbf{S}^T}{c}##, but now I have problems showing -{## \sigma_{ij}##}. I assumed the following for the calculation: $$F^{\mu \sigma} F_{\ \sigma}^{\...

I just want to make sure I'm doing this right. I know how to do the rotation, but reflection isn't demonstrated in the text. From what I'm seeing in the book, it seems like you take the matrix and simply multiply it by ##\begin{bmatrix} x \\ y \end{bmatrix} ## Assuming that's the case, I get...

I need to implement a routine for finding the eigenvalues of a symmetrical matrix (for computing the vibrational frequencies from Hessian). I had already implemented a Jacobi diagonalization algorithm, and in most cases it works properly, but sometimes it crashes. In particular, it crashes when...

Hi i have problems, to solve task a) Since I have to calculate the trace of the matrix ##Q##, I started as follows: $$\text{trace} (Q)=\sum\limits_{i=1}^{3}\int_{}^{}d^3x'(3x_i^{'2}-r^{'2}) \rho(x')$$ I then calculated further until I got the following form: $$\text{trace}...

TL;DR Summary: I have to find a system of equations with this solution ## {(1,2,0,3)^T+t(1,1,1,-2)^T+s(1,-1,3,0)^T;s,t \in \mathbb{R}} ## when we know that matrix of this equation has: 1. two non-zero rows 2. 3 non-zero rows. My idea is that I could somehow use the fact that...

I've seen dot product being represented as a (nx1 vector times a (mx1)^T vector. This gives a 1x1 matrix, whereas the dot product should give a scalar. I have found some threads online saying that a 1x1 matrix IS a scalar. But none of them seem to answer this question: you can multiply a 2x2...

The title pretty much says it... I know that in general eigenvalues are not necessarily preserved when matrix rows or columns are swapped. But in many cases it seems they are, at least with 4x4 matrices. So is there some specific rule that says when eigenvalues are preserved if I swap two rows...

In Dirac's "General Theory of Relativity", Chap. 34 on the polarization of gravitational waves, he introduces a rotation operator ##R##, which appears to be a simple ##\pi/2## rotation, since $$R \begin{pmatrix} A_0 \\ A_1 \\ A_2 \\ A_3 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 &...

I need to compute the vibrational frequencies of a molecule when the matrix of force constants (second derivative of the energy by the Cartesian coordinates) is provided. For such computation, this matrix must be diagonalized. Here is an example of a matrix which must be diagonalized: Here...

For example, consider the following system of 2 first order ODEs: $$ \left\{\begin{array}{l} x_1^{\prime}=2 t x_1+t^2 x_2 \\ x_2^{\prime}=t^3 x_1+4 t x_2 \end{array}\right. $$ This is a linear homogeneous system of 2 first order ODEs with $$A(t)=\left[\begin{array}{ll}2 t & t^2 \\ t^3 & 4...

I didn't have any good way to put this in the homework statement, but this is what the question is asking: For what c-value(s) will there be a non-trivial solution: ## x_1 - x_2 + x_3 = 0 ## ## 2x_1 + x_2 + x_3 = 0 ## ##-x_1 + (c)x_2 + 2x_3 = 0 ## I have spent a good couple hours looking at...

How does one find the dual of a matrix? Thanks.

Currently reading a textbook on non-equilibrium green's functions and I'm stuck in chapter 1 where it recaps just general quantum mechanics because of dirac deltas included in the matrix elements of a generalized hamiltonian. The textbook gives this: I just don't understand how to think about...

My working is , Consider case where the there are two linearly independent solutions ##x'(t) = c_1x' + c_2y' = A(c_1x + c_2y)## ##(x'~y')(c_1~c_2)^T = A(x~y)(c_1~c_2)^T## Then cancelling coefficient matrix I get, ##(x'~y')= A(x~y)## ##Φ'(t) = AΦ(t) ## by definition of 2 x 2 fundamental matrix...

Hello All, 2nd year undergrad taking my first course in modern physics. We have been given this question in a mock exam and at the bottom is the solution. When looking at a general cubit it seems the argument of both sin and cos functions should be (pi/2) not (pi/3). I have figured out a...

For this problem, The solution is, However, can someone please explain to me where they got the orange coefficient matrix from?It seems different to the original system of the form ##\vec x' = A\vec x## which is confusing me. Thanks!

My attempt is: Condition for critical point is ##x' = y' = 0##, ##0 = x - 2y \implies 2y = x## ##-2x + dy = 0## Then ##-4y + 4y = 0## However, this means that critical points are ##(2y, y)## as system is linearly dependent (both equations are the same) where ##y \in \mathbb{R}##. However, that...

We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds. Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x## Now consider ##n = m## case, ##B^m\vec x = B(B^{m - 1}...

I have a doubt about this problem. (a) Show that a matrix ##\left(\begin{array}{ll}e & g \\ 0 & f\end{array}\right)## has determinant equal to the product of the elements on the leading diagonal. Can you generalize this idea to any ##n \times n## matrix? The first part is simple, it is just ef...

Hello. I know that a 3×3 orthogonal matrix with determinant = 1 (so a 3×3 special orthogonal matrix) is a rotation in 3D. I was wondering if there is a 3×3 orthogonal matrix with determinant = –1 could be visualised in some way. Thank you!

Hello everyone, A simple ring resonator with a bus waveguide is described by: $$ \begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ -k^* & t^* \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix} $$ I do not understand though why we have -k* and t*? Shouldn't...

For part (b) i was able to use equations to determine the eigenvectors; For example for ##λ =6## ##12x +5y -11z=0## ##8x-4z=0## ##32x+10y-26z=0## to give me the eigen vector, ##\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}## and so on. My question is to get matrix P does the arrangement of...

I hope this is more properly laid out? We previously established that the stationery points were (1,1) and (-1,1) For this first stage I now need to create the elements of a Jacobian maitrix using partial differentation. I am confused by reference to the chain rule. Am I correct that for dx/dt...

Please confirm or deny the correctness of my understanding about this definition. For a given set of ##t_i##s, the matrix ##(B(t_i,t_j))^k_{i,j=1}## is a constant ##k\times k## matrix, whose entries are given by ##B(t_i,t_j)## for each ##i## and ##j##. The the 'finite' in the last line of the...

I have never solved a matrix ODE before, and am wondering if solving it is similar to solving ##y'=ay## where ##a## is a constant and ##y:\mathbb{R} \longrightarrow \mathbb{R}## is a function. The solution is right according to wikipedia, and I am just looking for your inputs. Thanks...

Let X be a continuous-time Markov chain that hops between two states ##\{1, 2\}## with rates ##\lambda, \mu>0##, so its generator is $$Q = \begin{pmatrix} -\mu & \mu\\ \lambda & -\lambda \end{pmatrix}.$$ Solve ##\pi Q = 0## for the stationary distribution, and verify that...

I have the matrix relationship $$C = A^{-1} B^{-1} A B$$ I want to solve for ##A##, where ##A, B, C## are 4x4 homogeneous matrices, e.g. for ##A## the structure is $$A = \begin{pmatrix} R_A & \delta_A \\ 0 & 1 \end{pmatrix}, A^{-1} =\begin{pmatrix} R_A^\intercal & -R_A^\intercal\delta_A \\ 0 & 1...

I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?

I am struggling to rederive equations (61) and (62) from the following paper, namely I just want to understand how they evaluated terms like ##\alpha\epsilon\alpha^{T}## using (58). It seems like they don't explicitly solve for ##\alpha## right?

hi, we have learned that after modelling Lagrangian and extracting Feynman rules from it - we can find matrix element - from which decay width can be calculated - and than Branching ratio - my question is can we use some other way of calculatiing BR , or can we use our Lagrangain in our Euler...

Let ##M## be a real ##n \times n## matrix. If ##M + M^T## is positive definite, show that $$\det\left(\frac{M + M^T}{2}\right) \le \det M$$

Hello everyone. I have four thermometers which measure the temperature in four different positions. The data is distributed as a matrix, where each column is a sensor, and each row is a measurement. All measurements are made at exactly the same times, one measurement each hour. I have...

The trace of the sigma should be the same in both new and old basis. But I get a different one. Really appreciate for the help. I’ll put the screen shot in the comment part

Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij? Quite confused about understanding this paragraph, can anybody explain it more easily?

Suppose ##A## and ##B## are positive definite complex ##n \times n## matrices. Let ##M## be an arbitrary complex ##n \times n## matrix. Show that the block matrix ##\begin{pmatrix} A & M\\ M^* & B\end{pmatrix}## is positive definite if and only if ##M = A^{1/2}CB^{1/2}## for some matrix ##C## of...

It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they...

A blog post by Evan Chen https://blog.evanchen.cc/2016/07/12/the-structure-theorem-over-pids/ says that elementary row and column operations on a matrix can be interpreted as a change-of-basis. I assume this use of the phrase "change of-basis" refers to creating a matrix that uses a different...

Let ##A## be a complex nilpotent ##n\times n##-matrix. Show that there is a unique nilpotent solution to the quadratic equation ##X^2 + 2X = A## in ##M_n(\mathbb{C})##, and write the solution explicitly (that is, in terms of ##A##).

Every hermitian matrix is unitary diagonalizable. My question is it possible in some particular case to take hermitian matrix ##A## that is not diagonal and diagonalize it UAU=D but if ##U## is not matrix that consists of eigenvectors of matrix ##A##. ##D## is diagonal matrix.

The sigma tensor composed of the commutator of gamma matrices is said to be able to represent any anti-symmetric tensor. \sigma_{\mu\nu} = i/2 [\gamma_\mu,\gamma_\nu] However, it is not clear how one can arrive at something like the electromagnetic tensor. F_{\mu\nu} = a \bar{\psi}...

For this, What was wrong with the notation I used for showing that I has swapped the rows? The marker put a purple ? Any help greatly appreciated! Many thanks!

https://upload.wikimedia.org/wikipedia/commons/d/d1/Taxonomy_of_Complex_Matrices.svg

For this, I am confused by the second line. Does someone please know how it can it be true since the matrix dose not have an inverse. Many thanks!

I found a the answer in a script from a couple years ago. It says the kinetic energy is $$ T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2 $$ However, it doesn't show the rotation matrix ##R##. This would imply...

For this problem, Find ##A^{-1}## given, The solution is, However, in the first image, why are we allowed to put together the submatrices in random order? In general does someone please know why we are allowed to decompose matrices like this? Many thanks!

For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!

For this, Dose someone please know where they get P and D from? Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k? Many thanks!

The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...