Matrix Definition and 1000 Threads

I have some experience with non-linear least squares curve fitting. For instance, if I want to fit a Gaussian curve to a set of data, I would use a non-linear least squares technique. A "model" matrix is implemented and combined with the observed data. The solution is found by applying well...

Homework Statement For the choelsky method , i was told by my lecturer that all the leading diagonal a11 , a22 and a33 must be the same... But , when I tried to find online resources , I found that that it's not stated in the rule that the leading diagonal a11 , a22 and a33 must be the same...

Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE F'...

Hi, I am studying Matrix chain Multiplication to find out the optimal way of multiplying a series of matrices so that we can reduce the number of multiplications. I have got this example from the book which multiplies the matrices having dimensions given below: A1 30 * 35...

Hi, I'm trying to understand which mathematical actions I need to perform to be able to arrive at the solution shown in the uploaded picture. Thank you.

If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?

Hi! I have a question regarding making the transition matrix for the corresponding probabilities. The main problem I feel I have here is figuring out how to represent the probabilities in the question in the transition matrix. Like if something is 7 times more likely than something else.. Any...

What are the coordinates on the 3D Bloch ball of a qubit's mixed state of the form: ##\rho=p_{00}|0\rangle \langle 0|+p_{01}|0\rangle \langle 1|+p_{10}|1\rangle \langle 0|+p_{11}|1\rangle \langle 1|##

Anyone know how to get a vertical line between two columns ? Like here in AMSTeX

This is a somewhat vague question that stems from the entries in a directional cosine matrix and I believe the answer will either be much simpler or much more complicated than I expect. So consider the transformation of an arbitrary vector, v, in ℝ2 from one frame f = {x1 , x2} to a primed...

Hello everyone, I am currently trying to understand how we can use feynman diagrams to estimate the matrix element of a process to be used in fermi's golden rule so that we can estimate decay rates. I am trying to learn by going through solved examples, but I am struggling to follow the logic...

Consider the matrix ##C = \gamma^{0}\gamma^{2}##. It is easy to prove the relations $$C^{2}=1$$ $$C\gamma^{\mu}C = -(\gamma^{\mu})^{T}$$ in the chiral basis of the gamma matrices.1. Do the two identities hold in any arbitrary basis of the gamma matrices? 2. How is ##C## related to the charge...

Homework Statement Given a matrix M={{2,1},{1,2}} then value of cos( (π*M)/6 )Homework EquationsThe Attempt at a Solution Eigen values are π/6 and π/2 and eigen vectors are (π/6,{-1,1}) and (π/2,{1,1}). Diagonalize matrix is {{π/6,0},{0,π/2}} I got same value (√3/2)M

Homework Statement Suppose that ##A^2 = 0##. Show that ##A## is not an invertible matrix Homework EquationsThe Attempt at a Solution We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##, which is to say that A is not...

Homework Statement Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). Homework EquationsThe Attempt at a Solution So I know the properties that define a group are...

Homework Statement Show that every matrix A ∈ O(2, R) is of the form R(α) = cos α − sin α sin α cos α (this is the 2d rotation matrix -- I can't make it in matrix format) or JR(α). Interpret the maps x → R(α)x and x → JR(α)x for x ∈ R 2 Homework EquationsThe Attempt at a Solution So I know...

I am taking a linear algebra class, and it has a required lab associated with it. Here is the following problem that I must solve using Matlab 1. Homework Statement Write a function using row reduction to find the inverse for any given 2x2 matrix. Name your function your initial + inv(M), the...

Homework Statement When is a Markov chain with double stochastic matrix positive recurrent? Homework Equations Double stochastic matrix is when the sum of the column vectors, and not just the row vectors, is 1. The Attempt at a Solution I know I have to show that the expected value of the...

Homework Statement Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential. a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are...

Homework Statement Use the given info to find matrix B Homework Equations (I + 3B)^-1 = [5 2; 4 2] to make more clear: inv(I + 3B) = this 2x2 matrix: top row = 5 2, bottom row = 4 2 The Attempt at a Solution I tried multiplying both sides of the eqn by I + 3B to get I = [5 2; 4...

I want to find the orthogonal matrix ##\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}## which diagonalises the matrix ##\begin{pmatrix} 0 & m\\ m & M \end{pmatrix}##. The eigenvalues are easily found to be ##\lambda = \frac{M}{2} \pm...

Homework Statement $$ A = \begin{bmatrix} 1 & 2\\ 2 & h\\ = k \end{bmatrix} $$ Mod note: Corrected augmented matrix: ##\begin{bmatrix} 1 & 2 & | & 2 \\ 2 & h & | & k \end{bmatrix}## Homework EquationsThe Attempt at a Solution Ok, so apparently it's a bad idea to...

Homework Statement Build the matrix A associated with a linear transformation ƒ:ℝ3→ℝ3 that has the line x-4y=z=0 as its kernel. Homework Equations I don't see any relevant equation to be specified here . The Attempt at a Solution First of all, I tried to find a basis for the null space by...

Homework Statement Show $$\frac{\partial \det(A)}{\partial A_{pq}} = \frac{1}{2}\epsilon_{pjk}\epsilon_{qmn}A_{jm}A_{kn}$$ Homework Equations ##\det(A)=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}## The Attempt at a Solution $$\frac{\partial \det(A)}{\partial A_{pq}}=\frac{\partial}{\partial...

Homework Statement So in the attachment you'll see a picture taken from a linear algebra book where a linear system of equations is presented in the equivalent augmented matrix form. I'm confused about the representation of the first equation in the augmented matrix. What happened to the...

I split off this question from the thread here: https://www.physicsforums.com/threads/error-in-landau-lifshitz-mechanics.901356/ In that thread, I was told that a symmetric matrix ##\mathbf{A}## with real positive definite eigenvalues ##\{\lambda_i\} \in \mathbb{R}^+## is always real. I feel...

Homework Statement [/B] Given this matrix ##\begin{bmatrix}As+B \\ C \end{bmatrix}## which is invertible and ##A## has full row rank. I would like to show that its inverse has no terms with ##s## or higher degree if ##\begin{bmatrix}A \\ C \end{bmatrix}## is invertible. Homework Equations...

Hello, Just wondering if the trace of a matrix is independent of basis, seeing as the trace of a matrix is equal to the sun of the eigenvalues of the operator that the matrix is a representation of. Thank you

Consider an algebraic variety, X which is a smooth algebraic manifold specified as the zero set of a known polynomial. I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of X, or more precisely, of the Jacobian variety...

Homework Statement Use matrix multiplication to find the 2×2 matrix P which represents projection onto the line y =√3x. Can you suggest another way of finding this matrix? Which vectors x∈R2 satisfy the equation Px = x? For which x is Px = 0? Homework Equations Dot product of vectors The...

I am currently brushing on my linear algebra skills when i read this For any Matrix A 1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out ) 2)(A + At)/2 is symmetric Now my question is , why should it be...

Hello all again, A is a matrix with order nXn, such that: \[A^{3}-2A^{2}+I=0\] I need to choose the correct answer: 1) A is not invertible 2) It is not possible to say if A is invertible 3) \[(A^{-1})^{2}=2I-A\] 4) \[A^{-1}=2I-A\] I can't find the solution here. I tried my own, and got...

Hello all I have this matrix: \[\begin{pmatrix} 6 & 0\\ -3 & a \end{pmatrix}\] And I am told it is diagonalizable. Therefore, the value of a is: 1) a=0 2) a not= 0 3) a not=6 4) a=6 5) a not=0,6 How should I approach this? Is there a "trick" or should I find eigenvalues and eigenvectors for...

Hello! I don't know exactly how to state my question so I'll show you what my problem is. Ex. Let T : R[x]_3 →R be the function defined by T(p(x)) = p(−1) + \int_{0}^{1} p(x) \,dx , where R[x]_3 is a vector space of polynomials with degree at most 3. Show that $T$ is a linear map; write down...

Let K be any Matrix, not necessarily the hamitonian. Is $$e^{-Kt}\left|\psi\right>$$ equal to $$e^{-K\left|\psi\right>t}$$ even if it is not the the eigenvector of K? I think so as i just taylor expand the $$e^{-Kt}$$ out but I want to confirm. In that case can i say that...

If you have a density matrix \rho, there is a basis |\psi_j\rangle such that \rho is diagonal in that basis. What are the conditions on \rho such that the basis that diagonalizes it is unique? It's easy enough to work out the answer in the simplest case, of a two-dimensional basis: Then \rho...

Homework Statement Homework Equations N/A The Attempt at a Solution I know that they got a rank of 2 since there are 2 linearly independent columns but what if we decided to count rows? In that case we would have 4 linearly independent rows which would suggest the rank is 4? How do we...

Suppose we have a product formed by a multiplication of a unitary matrix U and a diagonal matrix A, can we retrieve the inverse of A without knowing either U or A?

Hi! I'm studying Lie Algebras and Lie Groups. I'm using Brian Hall's book, which focuses on matrix lie groups for a start, and I'm loving it. However, I'm really having a hard time connecting what he does with what physicists do (which I never really understood)... Here goes one of my questions...

In Schwartz's book, 'Quantum Field Theory and the Standard Model' P.696, he starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. Most of the derivation is clear to me, except a couple of things at the start and midway. The...

Hello, suppose I have four conductors (1,2,3,4) and I know their mutual capacitances cij where i,j∈{1,2,3,4}. Note that the quantities cij are essentially the elements of the capacitance matrix of this system. Now, if I apply a voltage to two conductors and leave the other two grounded (e.g...

can anyone help me ? i have a homework and i did't find any answer for it the question is find the Domain , Range , matrix and the digraph for the relation R a ) A = { 1,2,3,4,8 } = B , aRb if and only if a=b b) A = { 1,2,3,4,6 } =B , aRb if and only if a multiple of b

Hey! :o I want to show that for $A,B\in \mathbb{R}^{2\times 2}$ the $U=\{X\in \mathbb{R}^{2\times 2}\mid AX=XB\}$ is a vector subspace of $\mathbb{R}^{2\times 2}$. We have that it is non-empty, since the zero matrix belongs to $U$ : $AO=O=OB$. Let $X_1, X_2\in U$ then $AX_1=X_1B$ and...

Is there any shortcut to find the rank of this $4 \times 6$ matrix quickly? $$A = \begin{pmatrix} -3 &2 &-1 &-2 &7 &-1\\ 9 &2 &27 &18 &7 &-9\\ 3 &2 &1 &0 &7 &-1\\ 6 &2 &8 &4 &-7 &-4\\ \end{pmatrix}$$ The above is a sample question for semester final test. If it were a homework, of course I...

Homework Statement T/F: The matrix ##\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}## is diagonalizable. Homework EquationsThe Attempt at a Solution Is there a quick way to tell whether the matrix is diagonalizable? Since it's a T/F question, that would seem to...

Homework Statement T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1 Homework EquationsThe Attempt at a Solution I know that if A is invertible and ##A\vec{v} = \lambda \vec{v}##, then ##A^{-1} \vec{v} = \frac{1}{\lambda} \vec{v}##, which seems to imply that A and...

Homework Statement A determinant with all elements of order unity may be surprisingly small. The Gilbert determinant Hij=(I+j-1)^-1, i,j=2... n is notorious for its small values. Homework EquationsThe Attempt at a Solution I just need help setting up the matrix and I can solve it myself. Thanks

i All, I have a Jupyter Python Notebook with data like below: \ I want to create an SFrame with 2 columns and 11 rows.Each row has two entries: One containing the name of each word and the other entry containing the total count of the word. The words are part of a list called 'Selected...

Homework Statement Hi everybody! While doing some homework for school, I realized that I still struggle to get what are the elements of an optical system matrix referring to. Here is the problem: An optical tube with length ##L=50##cm has at one end a convex lens (##D=2##) and at the other...

I'm reading this paper. But I haven't read anything on how to calculate the density operator in a QFT or how to calculate its trace. Now I can't follow this part of the paper. Can anyone clarify? Thanks