Matrix Definition and 1000 Threads

I have a matrix, [ a, ib; -1 1] where a and b are constants. I have to represent and analyse this matrix in a Hilbert space: I take the space C^2 of this matrix is Hilbert space. Is it sufficient to generate the inner product: <x,y> = a*ib -1 and obtain the norm by: \begin{equation}...

Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the...

Homework Statement Show that if ##λ##and ##V ## are a pair of eigenvalue and eigenvector for matrix A, $$e^Av=e^λv$$ Homework Equations ##e^A=\sum\limits_{n=0}^\infty\frac{1}{n!}A^n## The Attempt at a Solution I don't know where to start.

Homework Statement Let ##T:ℝ^3→ℝ^2## be the linear transformation defined by ##\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\mapsto \begin{bmatrix} x_1 + x_2 + x_3\\ 0 \end{bmatrix}##. i. Find the standard matrix for ##T##. Homework EquationsThe Attempt at a Solution For this problem I was...

Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...

Homework Statement (i) Reduce the system to echelon form C|d (ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent. (iii) Repeat part (b) above for k = −18 Homework Equations Gaussian elimination I used here...

Hi, I have the following ODE: aY'' + bY' + c = 0 I would like to convert it to a matrix, so to evaluate its eigenvalues and eigenvectors. I have done so for phase.plane system before, however there were two ODEs there. In this case, there is only one, so how does this look like in a matrix...

Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...

Homework Statement Show that if ##A^2## is the zero matrix, then the only eigenvalue of ##A## is 0. Homework Equations ##Ax=λx##. The Attempt at a Solution For ##A^2## to be the zero matrix it looks like: ##A^2 = AA=A[A_1, A_2, A_3, ...] = [a_{11}a_{11}+a_{12}a_{21}+a_{13}a_{31} + ... = 0...

Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...

Homework Statement Given this matrix: I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down. Homework Equations Gaussian elimination methods I used given here...

Hi, I have a matrix of an ODE which yields complex eigenvalues and eigenvectors. It is therefore not Hermitian. How can I further analyse the properties of the matrix in a Hilbert space? The idea is to reveal the properties of stability and instability of the matrix. D_2 and D_1 are the second...

Homework Statement Consider an n x n matrix A with the property that the row sums all equal the same number S. Show that S is an eigenvalue of A. [Hint: Find an eigenvector.] Homework Equations ##Ax=λx## The Attempt at a Solution S is just lambda here, so I begin solving this just like you...

Homework Statement Find the eigenvalues of the matrix ##\begin{bmatrix} 4 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & -3 \end{bmatrix}## Homework Equations ##Ax=λx## The Attempt at a Solution I'm having some trouble finding the eigenvalues of this matrix. The eigenvalue of a matrix is a scalar λ such...

Homework Statement Then solve these equations for i1-4 Homework Equations V=IR The Attempt at a Solution 80i_1-50i_2-30i_3 = -120 -50i_1+100i_2-10i_3-25i_4 = 0. -30i_1-10i_2+65i_3-20i_4 = 0. -25i_2-20i_3+100i_4 = 0. i_1=-4.18239492 i_2=-2.66455194 i_3=-2.71213323 i_4=-1.20856463...

Homework Statement Let ##A## be a 2x3 matrix. If Nul(##A##) is a line through the origin in ℝ3, then Col(##A##) = ℝ2. Explain why. Hint: Think about the number of pivots in ##A##. Homework EquationsThe Attempt at a Solution So, Nul(##A##) is the set of all solutions to the equation ##Ax=0##...

Homework Statement Coupled Harmonic Oscillators. In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 6.2 to the case where the oscillators are coupled. Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure...

Homework Statement Solve the following coupled differential equations by finding the eigenvectors and eigenvalues of the matrix and using it to calculate the matrix exponent: $$\frac{df}{dz}=i\delta f(z)+i\kappa b(z)$$ $$\frac{db}{dz}=-i\delta b(z)-i\kappa f(z)$$ In matrix form...

We have ##B=(1, X, X^2, X^3)## as a base of ##R3 [X]## and we have the endomorphisms ##d/dX## and ##d^2/dX^2## so that: ##d/dX (P) = P'## and ##d^2/dX^2 (P) = P''##. Calculating the matrix in class, the teacher found the following matrix, call it ##A##: \begin{bmatrix} 0 & 1 & 0 & 0...

Homework Statement Homework EquationsThe Attempt at a Solution I keep getting something over 0 for my Y_11. I'm not sure what I'm doing wrong. I thought the ideal transformer does not have an admittance or impedance matrix, which is why I should be getting something over 0 but the there...

Given that the Hamiltonian is H = P^2/(2m) + aδ(X − x(naught)) + bδ(X + x(naught), where x(naught) is a positive number. Find the conditions for bound states to exist and calculate their energies. Find the scattering matrix for arbitrary values of a and b. Can someone help me solve this please.

<Moved from a homework forum. Template removed.> I can't find any documentation on how to do this. I remember in linear algebra how to find the incidence matrix of an electrical network of purely resistors. Put how do I find it of a RLC circuit with resistors, inductors, and capacitors? I can't...

My teacher told me to find the definition of matrix which is in function form, but haven't seen it. The definition of matrix that I know is a rectangular arrangement of mn numbers, in m rows and n columns and enclosed within a bracket, but it is not right which my teacher wants. I want to know...

I'm trying to get the hang of index gymnastics, but I think I'm confused about the relationship between rank-2 tensor components and their matrix representations. So in Hartle's book Gravity, there's Example 20.7 on p. 428. We're given the following metric: ##g_{AB} = \begin{bmatrix} F & 1 \\...

Hi Lets say I have a txt file with m rows with n columns of numbers of the form: Lets say I want to take every p'th row and take the second and third columns and turn it into a \frac{m}{p}\times 2 matrix. How would I go about doing that?

I want a QM textbook which introduces detail knowledge of density matrix and trace (i.e. the average), who can recommend one for me? Thank you.

Homework Statement given that X is an n × p matrix with linearly independent columns. And $$X^∗ = XA$$ where A is an invertible p × p matrix. a) Show that: $$X^*{({X^*}^TX^*)^-}^1{X^*}^T = X{(X^TX)^-}^1X^T$$ b) Consider two alternative models $$M : Y = Xβ + ε$$ and $$M^∗ : Y = X^∗β ^∗ +...

How we can be sure that we are not living in matrix kind of virtual reality where we even do not have our bodies but all we have is our brain kept in jar of some liquid ? then also how we can be sure that the history as we know it till the last microsecond is totally made up and has been...

SOLVED Hi there. I have the elements of a matrix written in a txt file (in row major order). I need to read this matrix to use it in my fortran77 program. The text file contains the elements written in this way: A(1,1) A(1,2) ... A(1,N) ... A(N,N-1) A(N,N). I was thinking in doing a do loop...

Say we have a matrix L that maps vector components from an unprimed basis to a rotated primed basis according to the rule x'_{i} = L_{ij} x_{j}. x'_i is the ith component in the primed basis and x_{j} the j th component in the original unprimed basis. Now x'_{i} = \overline{e}'_i. \overline{x} =...

Hi All, I am trying to see if there is a "nice" ( relatively straightforward) way of finding the solution/kernel of the map : ##f(A)=A^n -Id ## , where A is an ## k \times k ## matrix and ##n## is a positive integer. Question: what is the kernel of this map? Cranking out matrix coefficients...

Homework Statement Need to prove that the derivative of a rotation matrix is a skew symmetric matrix muktiplied by that rotation matrix. Specifically applying it on the Rodrigues’ formula.Homework EquationsThe Attempt at a Solution I have shown that the cubed of the skew symmetric matrix is...

I want to transform the first matrix below into the second one. The book ( Neutsch, "Coordinates") says this can be done by elementary transformation. I guess he means by some Gaussian elimination. But the entries of the matrix are from the finite field 2, so I can not multiply rows, that would...

Homework Statement Homework Equations Matrix multiplication. The Attempt at a Solution Answer given=4 What am I doing wrong??

Hi, I'm trying to calculate the eigenvectors of a 4x4 matrix, but I don't want the actual eigenvalues included in the solution, I simply want them listed as a variable. For example, I have the matrix: H_F = \left[ \begin{array}{cccc} \hbar\Omega&\hbar v_fk_- &0&0\\ \hbar...

Homework Statement The question asks us to write down the matrix represented by the following summation. 2. Homework Equations The question summation... $$\sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k}$$ The Attempt at a Solution $$ \sum_{j,k=1}^{3} a_{ij}b_{jk}x_{k} =...

It is the demonstration of an important theorem I do not succeed in understanding. "A matrix has rank k if - and only if - it has k rows - and k columns - linearly independent, whilst each one of the remaining rows - and columns - is a linear combination of the k preceding ones". Let's suppose...

Hello Basically i need some help or references on proving that Working with spherical tensors in a 0_ ---> 0+ forbidden beta decay could you please give me some hints on how to do this proof? Thank you

Hey, I want to ask you about the array led component: 1.9mm(0.8’’)8*8 blue dot matrix led displays (model no:KWM-20882XBA). << Edit by Mentor to add link to datasheet >> https://cdn-shop.adafruit.com/datasheets/956datasheet.pdf I saw the technical data sheet but I couldn’t find the...

Hello. I fail to follow one step in the process of calculating ⟨la∥Y(L)∥lb⟩ . The spherical harmonics Yma(L)(r) represent the 2L+1 components of the spherical tensor of rank L. Writing the Eckart-Wigner th. for M = 0 yields: (1) Also one can write (2) Coupling L and lb to l: (3) Thus...

Homework Statement [/B] The trace of a matrix is defined to be the sum of its diaganol matrix elements. 1. Show that Tr(ΩΛ) = Tr(ΩΛ) 2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic) my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U...

Hi PF! Just want to make sure I'm not crazy: if we're solving a system ##K a = \sigma^2 M a## where ##K## and ##M## are ##n\times n## matrices, ##a## an ##n\times 1## vector and ##\sigma## a scalar, then ##a## is unnecessary, and all we really need to solve is ##K=\sigma^2 M##, right?

Hi. I have a real tridiagonal symmetric matrix that comes from the discretization of a partial differential equation. The elements are given by: ##A_{i,j}=-\delta_{i-1,j}\kappa_{i-1/2,l}\frac{\Delta t}{h^2}+\delta_{i,j}\left[\frac{2}{c}+\frac{\Delta t}{2}\mu_{i,j}+\frac{\Delta...

Hi there. I wanted to ask this question, which is about efficiency in matrix times vector multiplication in fortran. When I have some matrix ##\hat A## and vector ##\vec{x}##, and I want to compute the matrix times vector ##\hat A \vec{x}=\vec{b}## in Fortran, what I do is, I build the array...

Hi, I'm kind of stuck with this theorem stating that: if A is an unipotent matrix, then exp(log A) = A and also if X is nilpotent then log(exp X) = X Does anyone know any good approaches to prove this? I know that for unipotent A, logA will be nilpotent and that for nilpotent X, exp(X)...

I'm trying to construct a Fisher Forecast for the upcoming S4 CMB survey. I don't understand what the C_l is in this formula. It is H(z) and the Angular Distance? Or is it some covariance matrix and if it is a covariance matrix how do I calculate it considering the experiment hasn't been done...

I'm having trouble understanding a step in a proof about bilinear forms Let ## \mathbb{F}:\,\mathbb{R}^{n}\times\mathbb{R}^{n}\to \mathbb{R}## be a bilinear functional. ##x,y\in\mathbb{R}^{n}## ##x=\sum\limits^{n}_{i=0}\,x_{i}e_{i}## ##y=\sum\limits^{n}_{j=0}\;y_{j}e_{j}##...

Homework Statement Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\ c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc## Homework Equations N/A The Attempt at a Solution I got stuck at the first part. if ##A## is positive...

We use A = I.A as equation and then by transforming only A of LHS and I of RHS we come to I = P.A and we say that P is the inverse of matrix A My question is that why we only tranform A and I, why A of RHS is left as it is during the transformation, or why transformation do not take place in...

Homework Statement Find the transition matrix ##P## of a transformation defined as ##T:ℝ_2→ℝ_3## ##T:\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}a+2b\\-a\\b\end{bmatrix}## For basis ##B=\begin{bmatrix}1\\2\end{bmatrix},\begin{bmatrix}3\\-1\end{bmatrix}##...