Matrix Definition and 1000 Threads

As we know that 2×3 = 2+2+2 = 6; so similarly what does matrix multiplication represents?

The Internet is full of different communities, one such community is the 'Glitch in the Matrix' community, it is a big community and even has its own reddit page. People here discuss glitches they have experienced during the day, hundreds of people post everyday. Some posts are things that would...

Hello everyone. Iam working on a course in multivariable control theory and I stumbled over the Identity Matrix. I understand what the identity matrix is, though the use of it is a mistery... I was reading about going from state space to transfer functions and I found this expressions...

Homework Statement About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know. $$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$ Find the a) Jordan canonical form of ##A## b) characteristic polynomial c) minimal polynomial d) ##dim\,kerA## When: case 1: we know that ##A## is...

Homework Statement Let T: ℝ² → P² a linear transformation with usual operations such as T [1 1] = 1 - 2x and T [3 -1]= x+2x² Find T [-7 9] and T [a b] **Though I'm writing them here as 1x 2 row vectors , all T's are actually 2x1 column vectors (I didn't see a way to give them proper...

I posted this elsewhere and was sort of able to figure out a result myself, but 1) I didn't do it right, and 2) No one answered it anyway. I thought I'd give it a shot over here. The problem deals with nested matrices. The gamma matrices can be found here. My question deals with a "vector"...

##\begin{align}[A(BC)]_{ij} &= \sum_r A_{ir}(BC)_{rj} \\ &= \sum_r A_{ir} \sum_s B_{rs}C_{sj}\\ &= \sum_r\sum_s A_{ir}B_{rs}C_{sj}\\ &= \sum_{s} (\sum_{r} A_{ir} B_{rs}) C_{sj} \\ &= [(AB) C]_{ij}\end{align}## How did it went from ##2## to ##3##. In general is there a proof that sums can be...

If ##A## is ##m \times n## matrix, ##B## is an ##n \times m## matrix and ##n < m##. Then show that ##AB## is not invertible. Let ##R## be the reduced echelon form of ##AB## and let ##AB## be invertible. ##I = P(AB)## where ##P## is some invertible matrix. Since ##n < m## and ##B## is ##n...

Homework Statement Let ##A## be an ##m \times n## matrix. Show that by means of a finite number of elementary row/column operations ##A## can be reduced to both "row reduced echelon" and "column reduced echelon" matrix ##R##. i.e ##R_{ij} = 0## if ##i \ne j##, ##R_{ii} = 1 ##, ##1 \le i \le...

Homework Statement Find the inverse of ##A = \begin{bmatrix} 1 & \dfrac12 & & \cdots && \dfrac1n \\\dfrac12 & \dfrac13 && \cdots && \dfrac1{n+1} \\ \vdots & \vdots && && \vdots \\ \dfrac1n & \dfrac1{n+1} && \cdots && \dfrac1{2n-1}\end{bmatrix}## Homework EquationsThe Attempt at a SolutionI...

Hello friends, I'm trying to construct transformation matrix S such that it transforms Dirac representations of gamma matrices into Chiral ones. I know that this S should be hermitian and unitary and from this I arrived an equation with 2 matrices on the LHS (a known matrix multiplied by S from...

Prove that interchange of two rows of a matrix can be accomplished by a finite sequence of elemenatary row operations of the other two types. My proof :- If ##A_k## is to be interchanged by ##A_l## then, ##\displaystyle \begin{align} A_k &\to A_l + A_k \\ A_l &\to - A_l \\ A_l &\to A_k + A_l...

Hello, if I have some given vector c \in R^n, then I want to find solutions X \in R^{n\times n} to the following equation: X C X^T = C where C = c c^T. Certainly X = I is a solution, but I'm looking for any nontrivial solutions. We can also assume X is invertible if that helps. This equation...

I got an image called img and I want to sharpen the vertical spatial lines. I created this matrix: hp = [-3 -2 -1 12 -1 -2 -3]; hp= rot90(h_lowp); Then I do this to get the new image: newimg = uint8(conv2(single(hp), single(img))); It SORTA works, but I'm not so sure why it works. I mean all...

Suppose I have some arbitrary square matrix M, and I want to build a unitary matrix U: U=\left[\begin{array}{c|c}M & N \\\hline O & P\end{array}\right] Does there exist some general procedure for determining N, O, and P given M?

I have known: (1) A Hamiltonian, say, H(k), where k is the crystal momentum. (2) An appropriate complete basis set {a_1,a_2,a_3…}. (3) Some symmetric operators {A,B,…} which commute with H(k), i.e. [A,H]=[B,H]=...=0. Of course, by calculation, I can get any matrix element of H(k), i.e...

Homework Statement Using the following information, find the matrix A (I+2A)-1 = [-1 2] [4 5] Homework Equations AA-1 = I The Attempt at a Solution none. I have no idea how should I start. The inverse on the whole left side is driving me crazy.

Hi there. The question I wanted to ask is: Why are matrix methods so widely used for numerical solution of partial differential equations? Many times I've found that storing a whole matrix requires much more memory than just doing an iteration scheme to propagate the solution. Sometimes I...

Why is the dot product equivalent to the matrix multiplication of its components? I've seen some proofs using Pythagorean and cosine law but they don't give you an intuitive feel as to why matrix multiplication works. The geometric definition (##ab cosθ##) is very easy to understand. To a...

Homework Statement Define a simple random walk Yn on a finite state space S = {0, 1, 2, . . . , N} to be a random process that • increases by 1, when possible, with probability p, • decreases by 1, when possible, with probability 1 − p, and • remains unchanged otherwise. (a) Specify the...

Homework Statement [/B] The population is 50 The diseases is known to follow SIS dynamics with the following probabilities The number of infected individuals increases with probability 0.1 and it decreases with probability 0.05 the probability that nothing happens is 0.85 a) what is the...

Homework Statement If ##A## is an ##n \times n## nilpotent matrix, then the characteristic polynomial of ##A## is ##x^n## Homework EquationsThe Attempt at a Solution Suppose that ##A## has an eigenvalue with corresponding eigenvector such that ##A v = \lambda v##. Then ##A^k v = \lambda^k v =...

If given a position vector defined for a orthogonal curvilinear coordinate system HOW would the matrices that make up the Levi Civita 3x3x3 matrix remain the same? "Levi Civita 3x3x3 is said to be independent of any coordinate system or metric...

Homework Statement Create two 5x5 arrays, A & B, and ask the person to fill them out. Save those numbers in matrix_a.txt & matrix_b.txt respectively. Then, save the sum and difference of those numbers in sum.txt & diff.txt respectively. Basically we need to create two arrays, fill them out...

Homework Statement Hi guys, I am having an issue understanding what to do with this question. The question is displayed below: I have hand wirtten my working, as I don't now how to do matrices fully on latext. I used the definition to get this far for part a, but not sure about the second...

So the question is show that $$S=\left\{ \begin{pmatrix} a & b\\ -b & a \end{pmatrix} :a,b \in \Bbb{R} ,\text{ not both zero}\right\}$$ is isomorphic to $\Bbb{C}^*$, which is a non-zero complex number considered as a group under multiplication So I've shown that it is a group homomorphism by...

Homework Statement Find all components of the matrix eiaB. a is a constant and B is a 3x3 matrix whose first row is 0,0,-i second row is 0,0,0 and third row is i,0,0. The taylor expansion of eiaB gives 1+iaB-a2B2/2! - ... Homework Equations The taylor expansion of eiaB gives 1+iaB-a2B2/2! -...

Homework Statement 3.For which values of ##\lambda## does the following system of equations also have non trivial solutions Homework EquationsThe Attempt at a Solution What I tried doing first is to put all variables on the same side and got ## v+y-\lambda*x=0\\ x+z-\lambda*y=0\\...

I have a set of k-points, e.g. k1,k2,k3,k4. and they are equivalent by symmetry. Now I have calculated the momentum matrix element <i|p|j> at k1 point ONLY, and then calculate the optical properties which, for example, depend on <i|p|j><j|p|i> I have to make a summation on four k-points...

I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...

Hi. I must prove that, in general, the following relation is valid for the elements of a density matrix \rho_{ii}\rho_{jj} \geq |\rho_{ij}|^{2}. I did it for a 2x2 matrix. The density matrix is given by \rho = \left[ \begin{array}{cc} \rho_{11} & \rho_{12} \\ \rho^{\ast}_{12} & \rho_{22}...

I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...

Hi, got a question I'm stuck on.. Write down a matrix P which will diagonalise A and write down the corresponding diagonal matrix D, where D = P^-􀀀1AP. You do not have to calculate P^-1 Ive got all the eigenvalues and eigenvectors for A, and thus have the Matrix P, which has a determinant of...

This is a problem from Lang's Introduction to Linear Algebra. The problem statement is: Find a 2 x 2 matrix A such that A2= ##\begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix}## = -I The solution is available in the answer section of the book, but it is not shown how the author comes up with...

Homework Statement Let ##A## be an ##m \times n## matrix with rank ##m##. Prove that there exists an ##n \times m## matrix ##B## such that ##AB= I_m## Homework EquationsThe Attempt at a Solution So here is how far I get. I am given that ##A## has rank ##m##. Since ##L_A(x) = Ax## is a map...

Homework Statement Prove that if rank(A) = 0, then A = 0. Homework EquationsThe Attempt at a Solution This seems like a very easy problem, but I just want to make sure I am not missing anything. rank(A) = dim(Im(A)) = 0, so Im(A) = {0}. Thus, A is by definition the zero matrix. My only...

Homework Statement Find the Jordan canonical form of the matrix ## \left( \begin{array}{ccc} 1 & 1 \\ -1 & 3 \\ \end{array} \right)##. Homework EquationsThe Attempt at a Solution So my professor gave us the following procedure: 1. Find the eigenvalues for each matrix A. Your characteristic...

Hi, I have attached a pdf which shows clearly how I have carried out my transformations from one axis into another. However, I am not convinced that it is right and I have described why I feel so. I shall be grateful if someone can help me Kajal

Homework Statement Homework Equations ml2/12 The Attempt at a Solution So according to my databook: The axis x'-x' in the question corresponds to axis ZZ in the databook image above. That means in terms of radius, the moment of inertia about axis x'-x' is mr2/2. So in light of this, why...

Hello, What could be wrong when the total inertia matrix of a robotic manipulator is non invertible when under certain values of the joint angles? Thank you

How do we solve the particle in a box (infinite potential well) problem using matrix mechanics rather that using Schrodingers Equation? Schrodingers Equation for this particular problem is a simple partial differential equation and is easy for me to follow. The solution has the following...

Homework Statement how to find this Homework Equations none The Attempt at a Solution determinant??

Homework Statement Let ##B_1 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 0 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 0\\ 0 \end{bmatrix}}, {\begin{bmatrix} 0 \\ 0 \\ 1\\ 1 \end{bmatrix}} ## and ##B_2 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 1\\ -1 \end{bmatrix}}...

Homework Statement Write the density operator $$\rho=\frac{1}{3}|u><u|+\frac{2}{3}|v><v|+\frac{\sqrt{2}}{3}(|u><v|+|v><u|, \quad where <u|v>=0$$ In matrix form Homework Equations $$\rho=\sum_i p_i |\psi><\psi|$$ The Attempt at a Solution [/B] The two first factors ##\frac{1}{3}|u><u|##...

Consider the following tree-level Feynman diagrams for the ##W^{+}W^{-} \to W^{+}W^{-}## scattering process. The matrix element for this diagram can be read off from the associated quartic term ##\mathcal{L}_{WWWW}## in the electroweak boson self-interactions, where ##\mathcal{L}_{WWWW} =...

The system in which I tried to calculate the Hamiltonian matrix was a particle in a stadium (Billiard stadium). And I used the principle where we take a rectangle around the stadium in which the parts outside the stadium have a very high potential V0. We know the wave function of a rectangular...

Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns. The defining relation (necessary and sufficient) for Lorentz transforms is...

im wondering if anyone has any plausible theories that would disprove the possibility of the matrix? ive doe quite a few google searches and all i can find is the concept is quite possible. but surely there must be some ideas that would imply it can't be.

https://uploads.tapatalk-cdn.com/20170308/78feec183e9672f563c5e41b4c52e1d9.jpg https://uploads.tapatalk-cdn.com/20170308/4ad8560adf9e090969c38515a31d1407.jpg Please help, I know the definition of a cosine of a matrix is cos(a) = I-1/2!A^2+1/4!A^4-... But I am unsure how this would help me find...

Homework Statement We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a unique smooth solution F : I → gl(n;R), defined on the same interval I on which A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given.(i) Show that two solutions Fi : I →...