Matrix Definition and 1000 Threads

I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...

How can I make a curriculum to student, to explain the term Invertiable Matrix? What would I use to explain? [tools, whiteboard, papers. printed website pages...] How can I order the curriculum? What will be in the start of year and what will be next? This new material that I need to teach in...

So while I was trying to browse some notes with regard to Laplace transforms in solving systems of ODE, this matrix came up: I could easily just use RK4 and call it a day to nuke the systems of ODE, but this actually made me curious. Apparently one can transform the matrix DE into another form...

While the prefix of the thread is Python, this could be easily generalised to any language. It is absolutely not the first time I am working with an array, but definitely the first time I am facing the task of defining the transpose of a non-square matrix. I have worked so much with arrays in...

Show that ##\{\mathbf{v}_1\}## is linearly independent. Simple enough let's consider $$c_1\mathbf{v}_1 = \mathbf{0}.$$ Our goal is to show that ##c_1 = 0##. By the definition of eigenvalues and eigenvectors we have ##A\mathbf{v}_1= \lambda_1\mathbf{v}_1##. Let's multiply the above equation and...

Summary: What are the basic assumptions of QM about the density matrix? The subject of density matrix in quantum mechanics is very unclear to me. In the books I read (for example Sakurai),they don't tell what are the basic assumptions and how you derive from them the results of the density...

I have tried to do this using arrays and do loops: program matrixmul implicit none real A(2, 2), B (2, 2), C (2, 2) integer i, j, k write (*, *) 'Input: First matrix' do i = 1, 2 do j = 1, 2 read (*, *) A (i, j) enddo enddo write (*, *) 'Input: Second...

Show that a square matrix with a zero row is not invertible. first a matrix has to be a square to be invertable if $$\det \begin{pmatrix}1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}=1$$ then $$\begin{pmatrix} 1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}^{-1} =\begin{pmatrix}1&0&0\\ 0&1&0\\ -3&0&1...

I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...

I am fascinated again with The Matrix especially as it could pertain to the minimum instrumental version of quantum mechanics. Read this conversation (the quantum stuff added) The Matrix is also the minimum instrumental version of quantum mechanics which "explains" our daily world...

I'm trying to understand some notes that I have been given on Matrix Mechanics, specifically how the matrix element comes about and builds a matrix which when used applies the effect of an operator on a wavefunction. But I'm having some difficulties following what's being done in the notes with...

Hi PF! Given a matrix and vector $$ \begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix},\\ \begin{bmatrix} 1\\ 2 \end{bmatrix} $$ how can I merge the two to have something like this $$ \begin{bmatrix} (1,a) & (1,b) & (1,c)\\ (2,d) & (2,e) & (2,f) \end{bmatrix} $$

2000 5.1 Suppose that we know that $A^{-1}=\begin{bmatrix}1&3\\2&5 \end{bmatrix}$ Solve the matrix equation $AX=B$ to find $x$ and $y$ where $X=\begin{bmatrix}x\\y \end{bmatrix}\& \quad B=\begin{bmatrix}1\\3 \end{bmatrix}$ ok well first find A $A=\begin{bmatrix}1&3\\2&5 \end{bmatrix}^{-1}...

I have a simple interface between vacuum and a uniaxial crystal. While it was easy to determine the reflection using Fresnel's equations, the analysis needs to be done "using matrices". We are only interested in the TE/TM reflections. Which method works best for this? ABCD/Ray transfer matrix...

Hi, I've the following doubt: consider an homogeneous linear system ##Ax=0## with ##A## a singular square matrix. The resulting matrix attained through Gaussian elimination will be in upper triangular or raw echelon form ? Thanks.

The Hamiltonian of the system I'm working on is in the form : ##\hat H=\dfrac{p^2}{2m}-\dfrac{\partial_z^2}{2m}+V(z)+\gamma V'(z)(\hat z \times \vec{\mathbf p})\cdot \vec{\sigma}## There is translational symmetry in the x-y plane. ##\vec{\mathbf{p}}## is the two dimensional momentum in the x,y...

So if ##P_{0}## is an event, and I have ##\mathcal {g_{\mu\nu}(P_{0})}=0## and ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0##, does this notation mean ##\partial\alpha\partial\beta## or simply ##\partial(\alpha\beta)##? And what is the significance of it? Why can't it be zero in curved spacetime?

Look, I am sorry for not being able to post any LaTeX. But I am stuck at a place where I feel I should not be stuck. I can not figure out how to correctly do this. I can't seem to recreate the Pauli matrices with that form using the 3 2-dimensional bases representing x, y, and z spin up/down...

Hi, Please see the attached image. I have a matrix and would like to split it up into a nice compact equation if possible. Matrix A seems to be a nice pattern that would lend itself to writing in equation form but I’m not sure what to do. Is it possible? Also do you know how I could correctly...

I have a 4x4 operator O. I apply it on a 4x1 vector A. Let's say A =[0.7; 0.4 ; 0.4; 0.3]. When O acts on A, I get B. Let's say B=[0.74 ; 0.56; 0.08 ; 0.36]. The problem is I don't know how to find O. Can you please help me. My basis are [1 ; 0 ; 0; 0], [0;1 ; 0 ;0] ... and so on. Thanks...

Hi all, I've come across an interesting matrix identity in my work. I'll define the NxN matrix as S_{ij} = 2^{-(2N - i - j + 1)} \frac{(2N - i - j)!}{(N-i)!(N-j)!}. I find numerically that \sum_{i,j=1}^N S^{-1}_{ij} = 2N, (the sum is over the elements of the matrix inverse). In fact, I...

Let ## \begin{align}M =\begin{pmatrix} 2& -3& 0 \\ 3& -4& 0 \\ -2& 2& 1 \end{pmatrix} \end{align}. ## Here is how I think the JCF is found. STEP 1: Find the characteristic polynomial It's ## \chi(\lambda) = (\lambda + 1)^3 ## STEP 2: Make an AMGM table and write an integer partition...

Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.

Assume that ##\partial M_{ab}/\partial \hat{n}_c## is completely symmetric in ##a, b## and ##c##. Then, it is stated in the book I read that the divergence of the traceless part of ##M## is proportional to the gradient of the trace of ##M##. More precisely, $$ \partial /\partial \hat{n}_a...

The transformation matrix for a beam splitter relates the four E-fields involved as follows: $$ \left(\begin{array}{c} E_{1}\\ E_{2} \end{array}\right)=\left(\begin{array}{cc} T & R\\ R & T \end{array}\right)\left(\begin{array}{c} E_{3}\\ E_{4} \end{array}\right) \tag{1}$$ Here, the amplitude...

Hi. I'm learning Quantum Calculation. There is a section about controlled operations on multiple qubits. The textbook doesn't express explicitly but I can infer the following statement: If ##U## is a unitary matrix, and ##V^2=U##, then ## V^ \dagger V=V V ^ \dagger=I##. I had hard time...

Summary: I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space. I need to do a linear model identification using least squared method. My model to identify is a...

My first attempt was: V=zeros(5,5) a=1; i=1:5; j=1:5; V(i:j)=a./(i+j-1) I figured to create a 5x5 with zeros and then to return and replace those values with updated values derived from the Hilbert equation as we move through i and j. This failed with an error of : Unable to perform assignment...

Let A= [ a b] [ c d ] B = [ w x] [ y z] Then aw +by=0 bx+dz=0 cw+dy=0 cx+dz=0 aw+cx! =0 bw+x! =0 ya+cz!=0 by+dz! =0 But I don't get the answer after this

Is there a relationship between the momentum operator matrix elements and the following: <φ|dH/dkx|ψ> where kx is the Bloch wave number such that if I have the latter calculated for the x direction as a matrix, I can get the momentum operator matrix elements from it?

If a 3×3 matrix A produces 3 linearly independent eigenvectors then we can write them columnwise in a matrix P(non singular). Then the matrix D = P_inv*A*P is diagonal. Now for this I need to show that different eigenvalues of a matrix produce linearly independent eigenvectors. A*x = c1x A*y...

According to me matrix multiplication is not commutative. Therefore A^2.A^3=A^3.A^2 should be false. But at the same time matrix multiplication is associative so we can take whatever no. of A's we want to multiply i.e A^5=A.A^4 OR A^5=A^2.A^3

nmh{896} mnt{347.21} consider th non-homogeneous first order differential system where $x,y,z$ are all functions of the variable t \begin{align*}\displaystyle x'&=-4x-3y+3z\\ y'&=3x+2y-3z+e^t\\ z´&=-3x-3y+2z \end{align*} write a system in the matrix form $Y'=AY+G$

I was drawing out the multiplication table in "matrix" form (a 12 by 12 matrix) for a friend trying to pass the GED (yes, sad, I know) and noticed for the first time that the entries on the diagonal are real, i.e. the squares (1, 4, 9, 16, ...), and the off diagonal elements are real and complex...

Hello, Consider the system of linear homogeneous differential equations of first order dy/dx = A(x) y where x denotes the independent variable, A(x) is a square matrix, and y is an unknown vector-function...

2nd one is considerably hard to compute ##P^n## using simple matrix multiplication as the given matrix ##P## is cumbersome to work with. Also, I need to know how to test a matrix to find if that matrix has a limiting distribution. So, I need some help.

he is asking for the division of the two matrices , so i tried to get the inverse of the matrix A but it appears to get more complex as the delta for A is somehow a big equation . and what really bothers me that there is another A , B inside the matrix B ?! find B/A .

I just discovered this website and want to thank everyone who is willing to contribute some of their time to help me. I appreciate it more than you know First off, assume that state 1 is Chinese and that state 2 is Greek, and state 3 is Italian. A student never eats the same kind of food for 2...

In ##t = 0##, we have ##\rho (0) = | + \rangle \langle + |##. The time evolution of the density matrix is given by ##\rho(t) = e^{-i\hat{H}t} \rho (0) e^{i\hat{H}t}## (I am considering ##\hbar = 1##). I can write the state ##| + \rangle ## as a linear combination of the eigenstates of the...

Im following Weinberg's QFT volume I and I am tying to show that the following equation vanishes at large spatial distance of the possible particle clusters (pg 181 eq 4.3.8): S_{x_1'x_2'... , x_1 x_2}^C = \int d^3p_1' d^3p_2'...d^3p_1d^3p_2...S_{p_1'p_2'... , p_1 p_2}^C \times e^{i p_1' ...

I refer to the transition matrix for a Markov process and I have two questions 1. How can one tell if a Markov process is reversible ? 2. Can it have two (or more) eigenvalues equal to 1 ? My definition of the matrix is that it should have all rows(or columns) sum to 1. Thanks.

To see the steps I have completed so far, https://math.stackexchange.com/q/3168898/261956 I think there are at least three more steps. The next step is finding the eigenvectors together with the generalized eigenvectors of each eigenvalue. Then we use this to construct the transition matrix...

Hello This could very well be an idiotic question, but here goes... Consider a general matrix M Consider a rotation matrix R (member of SO(2) or SO(3)) Is it possible to split M into the product of a rotation matrix R and "something else," say, S? Such that: M = RS or the sum M = R + S...

Hello, I have some trouble understanding how to construct the matrix for the beam splitter (in a Mach-Zehnder interferometer). I started with deciding my input and output states for the photon. I then use Borns rule, which I have attached below: To get the following for the state space...

Homework Statement Diagonalize the matrix $$ \mathbf {M} = \begin{pmatrix} 1 & -\varphi /N\\ \varphi /N & 1\\ \end{pmatrix} $$ to obtain the matrix $$ \mathbf{M^{'}= SMS^{-1} }$$ Homework Equations First find the eigenvalues and eigenvectors of ##\mathbf{M}##, and then normalize the...

I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...

Homework Statement Hi, I am looking for an example to solve a larger Matrix by dividing into Quadrant. Is it possible for Gauss Elimination or Matrix Multiplication. Homework Equations No equation possible The Attempt at a Solution Looking for a example Zulfi.

Homework Statement The problem is to calculate the determinant of 3x3 Matrix by using elementary row operations. The matrix is: A = [1 0 1] [0 1 2] [1 1 0] Homework EquationsThe Attempt at a Solution By following the properties of determinants, I attempt to get a triangular matrix...