Matrix Definition and 1000 Threads

What would be reasonable preparation for reading the reference Kaon Physics https://arxiv.org/pdf/hep-ph/0401236.pdf I find I do not seem to be ready for this reading. I am familiar with quantum mechanics at the level of Schiff (an old text for first year graduate students) though perhaps rusty...

Good day All I have a doubt regarding the derivation of the following matrix according to my basic understanding we want to go from the basis u1 v1 u2 v2 to the basis u'1 v'1 u'2 v'2, and for doing so we use the rotation matrix the rotation matrix is the following and the angle theta is...

Good Day I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc. But I am making a lot of progress. There is, however, one issue that continues to elude me. I often read how Lie developed Lie Groups to study symmetries of PDE's May I ask if someone...

Homework Statement So I have these two Matrices: M = \begin{pmatrix} a & -a-b \\ 0 & a \\ \end{pmatrix} and N = \begin{pmatrix} c & 0 \\ d & -c \\ \end{pmatrix} Where a,b,c,d ∈ ℝ Find a base for M, N, M +N and M ∩ N. Homework Equations I know the 8 axioms about the vector spaces. The...

Homework Statement Let A be an arbitrary m× n matrix. Find a matrix C such that CA = B for each of the following matrices B. a. B is the matrix that results from multiplying row i of A by a nonzero number c. b. B is the matrix that results from swapping rows i and j of A. c. B is the matrix...

Homework Statement Hi good morning to all. The problem at hand states, that the points A (3,0) and B (5,0) are reflected in the mirror line y=x. Determine the images A' and B' of these points. I've done that using the reflection in the line y=x which i know to be \begin{bmatrix} 0 &1 \\ 1 & 0...

Homework Statement Homework Equations For Hermition: A = transpose of conjugate of A For Skew Hermition A = minus of transpose of conjugate of AThe Attempt at a Solution I think this answer is C. As Tranpose of conjugate of matrix is this matrix. Book answer is D. Am I wrong or is book wrong?

Hi, I am trying to find the eigenvectors for the following 3x3 matrix and are having trouble with it. The matrix is (I have a ; since I can't have a space between each column. Sorry): [20 ; -10 ; 0] [-10 ; 30 ; 0] [0 ; 0 ; 40] I’ve already...

Homework Statement PROBLEM STATEMENT: "Represent the grid patterns in the figure with a dither matrix." (Figure: https://www.docdroid.net/OMLUX5v/figure.pdf ) ANSWER (FROM MY BOOK): http://www.wolframalpha.com/input/?i=%7B%7B0,2%7D,%7B3,1%7D%7D Homework Equations...

Hey! :o At the block deflation it holds for a non-singular Matrix $S$ \begin{equation*}SAS^{-1}=\begin{pmatrix}C & D \\ O & B\end{pmatrix}\end{equation*} where $O$ is the zero matrix. It holds that $\sigma (A)=\sigma(B)\cup \sigma (C)$, where $\sigma (M)$ is the set of all eigenvalues of a...

Hey! :o Let A be a regular ($n\times n$)-Matrix, for which the Gauss algorithm is possible. If we choose as the right side $b$ the unit vectors $$e^{(1)}=(1, 0, \ldots , 0)^T, \ldots , e^{(n)}=(0, \ldots , 0, 1 )^T$$ and calculate the corresponding solutions $x^{(1)}, \ldots , x^{(n)}$ then...

Hey! :o Let $A \in\mathbb{R}^{n\times n}$, $n\geq 3$ be a matrix with $n+1$ elements $1$ and the remaining elements are $0$. I want to show that $\det (A)\in \{-1, 0, 1\}$ and each of these $3$ possible values can occur. Could you give me a hint how we could show that? I got stuck right now...

Hello all, Given the following matrix, \[A=\begin{pmatrix} 2 & 6\\ 1 & a \end{pmatrix}\] and given that \[\lambda =0\] is an eigenvalue of A, I am trying to determine that value of a. What I did, is to create the characteristic polynomial \[(\lambda -2)*(\lambda -a)+6=0\] and given...

Hello all, If A and B are both squared invertible matrices and A is also symmetric and: \[AB^{-1}AA^{T}=I\] Can I say that \[B=A^{3}\] ? In every iteration of the solution, I have multiplied both sides by a different matrix. At first by the inverse of A, then the inverse of the transpose...

Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...

Hi! I am currently working on this question about matrices and showing they are homomorphisms. I have done part (i), but on part (ii) I am confused as the matrix is mapping to a - I have never seen this before and I'm not sure how to approach it. I know that usually you would work out the...

Prove, that there is no $2 \times 2$ matrix, $S$, such that \[S^n= \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}\] for any integer $n \geq 2$.

I'm studying Newton Raphson Method in Load Flow Studies. Book has defined Jacobian Matrix and it's order as: N + Np - 1 N = Total Number of Buses Np = Number of P-Q Buses But in solved example they've used some other formula. I'm not sure if it's right. Shouldn't order be: N + Np - 1 N = 40 Np...

Homework Statement Homework EquationsThe Attempt at a Solution [/B] Det( ## e^A ## ) = ## e^{(trace A)} ## ## trace(A) = trace( SAS^{-1}) = 0 ## as trace is similiarity invariant. Det( ## e^A ## ) = 1 The answer is option (a). Is this correct? But in the question, it is not...

Homework Statement This is just the triple integral of an easy matrix problem. I just have no ideas what they got by the time they got to the integral of x. Homework Equations integral[/B]The Attempt at a Solution Somebody please prove me wrong. I got a matrix of constants by the time I got...

Hi! I have an orthonormal basis for vector space $V$, $\{u_1, u_2, ..., u_n\}$. If $(v_1, v_2, ..., v_n) = (u_1, u_2, ... u_n)A$ where $A$ is a real $n\times n$ matrix, how do I prove that $(v_1, v_2, ... v_n)$ is an orthonormal basis if and only if $A$ is an orthogonal matrix? Thanks!

Hi, I am trying to prove that the eigevalues, elements, eigenfunctions or/and eigenvectors of a matrix A form a Hilbert space. Can one apply the inner product formula : \begin{equation} \int x(t)\overline y(t) dt \end{equation} on the x and y coordinates of the eigenvectors [x_1,y_1] and...

Homework Statement I am having a issue with how my lecture has normalised the energy state in this question. I will post my working and I will print screen his solution to the given question below, we have the same answer but I am unsure to why he has used the ratio method. Q4. a), b), c)...

Homework Statement I have attached the question. Translated: Suppose T: R^4 -> R^4 is the image so that: ... Homework Equations So I did this question and my final answers were correct: 1. not surjective 2. not injective. My method of solving this question is completely different than the...

I am trying to normalize 4x4 matrix (g and f are functions): \begin{equation} G=\begin{matrix} (1-g^2) &0& 0& 0&\\ 0& (1+f^2)& (-g^2-f^2)& 0 \\ 0 &(-g^2-f^2)& (1+f^2)& 0 &\\ 0& 0& 0& (1-g^2) \end{matrix} \end{equation} It's a matrix that's in a research paper (which I don't have) which gives...

Homework Statement Homework EquationsThe Attempt at a Solution I solved it by calculating the eigen values by ##| A- \lambda |= 0 ##. This gave me ## \lambda _1 = 6.42, \lambda _2 = 0.387, \lambda_3 = -0.806##. So, the required answer is 42.02 , option (b). Is this correct? The matrix is...

The following matrix A is, \begin{equation} A= \begin{bmatrix} a+b-\sigma\cdot p & -x_1 \\ x_2 & a-b-\sigma\cdot p \end{bmatrix} \end{equation} The inversion of matrix A is, \begin{equation} A^{-1}= \frac{\begin{bmatrix} a-b-\sigma\cdot p & x_1 \\ -x_2 & a+b-\sigma\cdot p...

Homework Statement I've created code to crack a Hill Cipher (n=3). I'm unsure which cribs to try to crack a specific code. Would anyone mind posting ideas? The crib must be 9 letters in length. Homework EquationsThe Attempt at a Solution

Homework Statement Find the trace of a ##4\times 4## matrix ##\mathbb U=exp(\mathbb A)##, where $$\mathbb A = \begin {pmatrix} 0&0&0&{\frac {\pi}{4}}\\ 0&0&{\frac {\pi}{4}}&0\\ 0&{\frac {\pi}{4}}&0&0\\ {\frac {\pi}{4}}&0&0&0 \end {pmatrix}$$ Homework Equations $$e^{(\mathbb A)}=\mathbb P...

Hi, I have derived a matrix from a system of ODE, and the matrix looked pretty bad at first. Then recently, I tried the Gauss elimination, followed by the exponential application on the matrix (e^[A]) and after another Gauss elimination, it turned "down" to the Identity matrix. This is awfully...

Hi, the three main types of complex matrices are: 1. Hermitian, with only real eigenvalues 2. Skew-Hermitian , with only imaginary eigenvalues 3. Unitary, with only complex conjugates. Shouldn't there be a fourth type: 4. Non-unitary-non-hermitian, with one imaginary value (i.e. 3i) and a...

Hey! :o An octopus is trained to chosose from two objects A and B always the object A. Repeated training shows the octopus both objects, if the octopus chooses object A, he will be rewarded. The octopus can be in 3 levels of training: Level 1: He can not remember which object was rewarded...

I have calculated that a matrix has a Frobenius norm of 1.45, however I cannot find any text on the web that states whether this is an ill-posed or well-posed indication. Is there a rule for Frobenius norms that directly relates to well- and ill-posed matrices? Thanks

Hi, I have the following complex ODE: aY'' + ibY' = 0 and thought that it could be written as: [a, ib; -1, 1] Then the determinant of this matrix would give the form a + ib = 0 Is this correct and logically sound? Thanks!

Hi everybody, I'm writing some algebra classes in C++ , Now I'm implementing the modified sparse row matrix , I wrote all most all of the class, but I didn't find the way saving computing time to perform the product of two Modified sparse row matrix .. if you don't know it you can read in the...

Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?

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...