Matrix Definition and 1000 Threads

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 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

Homework Statement Hi all! I'm having trouble understanding the implementation of some derivatives in the expression (1) of this article: https://www.ncbi.nlm.nih.gov/pubmed/26248210 How do I implement ∑(ij) ∂ijw ? Thank you all in advance. Homework Equations w is a square matrix(120x120)...

Homework Statement Consider an ensemble of spin 1 systems (a mixed state made of the spin 1 system). The density matrix is now a 3x3 matrix. How many independent parameters are needed to characterize the density matrix? What must we know in addition to Sx, Sy and Sz to characterize the mixed...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of Bresar's Example 1.10 on a simple matrix ring over a division ring ... Example 1.10, including some...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of Bresar's Example 1.10 on a simple matrix ring over a division ring ... Example 1.10...

$A=\begin{bmatrix} 3&2\\ \end{bmatrix} B=\begin{bmatrix} 1\\ 2\end{bmatrix}$ Find the value of the matrix $AB$. The order of the first matrix is 1*2 The order of the second matrix is 2*1 Matrix AB should be 1*1 I am a bit struggling in determining the way...

I am facing some difficulties solving one of the questions we had in our previous exam. I am sorry for the bad translation , I hope this is clear. In each section, find all approppriate matrices 2x2 (if exists) , which implementing the given conditions: is an eigenvector of A with eigenvalue...

Hello everyone, I'have implemented a Maximum-Likelihood-Expectation-Maximization Algorithm in order to reconstruct a bild. let say, we have such a system Ax=b, where A is a complex matrix, b is a complex vector. A and b are known and we will iterately try to find the best x (which should be...

Homework Statement Homework Equations determinant is the product of the eigenvalues... so -1.1*2.3 = -2.53 det(a−1) = 1 / det(A), = (1/-2.53) =-.3952 The Attempt at a Solution If it's asking for a quality of its inverse, it must be invertible. I did what I showed above, but my answer was...

I have a Hamiltonian represented by a penta-diagonal matrix The first bands are directly adjascent to the diagonals. The other two bands are N places above and below the diagonal. Can anyone recommend an efficient algorithm or routine for finding the eigenvalues and eigenvectors?

Hello all Let ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX## and ##A \in M_{m,n}(\mathbb{K})## (I already proved that this function is linear) I want to prove that: A regular matrix ##\iff m_A## is an isomorphism. So, here is my approach. Can someone verify whether this is...

Hey! :o We have that a matrix $A$ is idempotent if it holds that $A^2=A$. We suppose that $X$ is a $m\times n$-matrix and that $(X^TX)^{-1}$ exists. I want to show that $A=I_m-X(X^TX)^{-1}X^T$ is idempotent. I have done the following: $$A^2 =A\cdot A=(I_m-X(X^TX)^{-1}X^T)\cdot...

Hello everybody, From a complete set of orthogonal basis vector ##|i\rangle## ##\in## Hilbert space (##i## = ##1## to ##n##), I construct and obtain a nondiagonal Hamiltonian matrix $$ \left( \begin{array}{cccccc} \langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\...

Homework Statement Find out for which values of ##\alpha## the system ##AX + \alpha X = (1, 1, 1, 1)^t## has solutions. $$A = \begin{pmatrix} 6 & 0 & -1 & 2 \\ 3 & 5 & -3 & 6 \\ -2 & 0 & 7 & -4 \\ 2 & 0 & 1 & 0 \end{pmatrix} X = \begin{pmatrix} x \\ y \\ z \\ t \end{pmatrix}$$ Homework...

Is there any way that i can convert a non-hermitian matrix to a hermitian matrix ?

Homework Statement [/B] Homework Equations N/A The Attempt at a Solution What I am confused about is where they got the (1/4)mR^2 + (1/12)ml^2 and (1/2)mR^2 from? I am guessing that these came from the integral of y'^2 + z'^2 and x'^2 +y'^2 but I don't understand how this happened exactly...

Homework Statement Consider a quantum mechanical system with three states. At each step a particular particle transitions from one state to a different state. Empirical data show that if the particle is in State 1, then it is 7 times more likely to go to State 2 at the next step than to State...

Homework Statement Consider the following matrix. A = 2 + 4i...1 + 5i 2 − 3i...2 + 3i Let B = A-1. Find b12 (i.e., find the entry in row 1, column 2 of A−1) Homework Equations A-1 = 1/(ad - cb)* [ d -b ] [ -c a ] <--imagine as 2x2 matrix with first row (d,-b) and second row...