Matrices Definition and 1000 Threads

Homework Statement Let q ∈ C^m have 2-norm of q =1. Then P = qq∗ is a projection matrix. (a) The matrix P has a singular value decomposition with U = [q|Q⊥] for some appropriate matrix Q⊥. What are the singular values of P? (b) Find an SVD of the projection matrix I − P = I − qq∗ . In...

Hello! Im currently teaching a math course on a upper secondary school, where we are doing some linear algebra (it is the last math course before the students continue for university). After a brief introduction I want to show some interesting applications of matrices. Applications which can be...

Homework Statement A magnetic data set is believed to be dominated by a strong periodic tidal signal of known tidal period\Omega The field strength F(t) is assumed to follow the relation: F=a+b\cos\Omega t + c\sin\Omega t If the data were evenly spaced in time, then Fourier analysis would...

i used to get pauli matrices by the following steps it uses the symmetry of a complex plane sphere i guess so..? however i can't get the 8 gell mann matrices please help ! method*: (x y) * (a b / c d ) = (x' y') use |x|^2 + |y|^2 = |x'|^2 + |y'|^2 and |x| = x * x(complex conjugate) this way...

Are All symmetric matrices with real number entires Hermitian? What about the other way around-are all Hermitian matrices symmetric?

I'm trying to learn some basic quantum mechanics, mostly from I a mathematical perspective. I am trying to understand this with quantum states as vectors in a Hilbert space, bipartite systems, the difference between superpositions and ensembles of states, and density matrices. And it is the two...

Hello, I was refreshing my Mathematics using S.M. Blinder's book "Guide to Essential Math" and on the section on Matrix Multiplication I got the following, Can someone elaborate on the highlighted section? In particular, what does "adjacent indices" mean? Thank you.

Consider the operator Ô, choose a convenient base and obtain the representation of \ exp{(iÔ)} Ô = \bigl(\begin{smallmatrix} 1 & \sqrt{3} \\ \sqrt{3} & -1 \end{smallmatrix}\bigr) Attempt at solution: So, i read on Cohen-Tannjoudji's Q.M. book that if the matrix is diagonal you can just...

Homework Statement Prove that the upper triangular matrices form a subspace of ##\mathbb{M}_{m \times n}## over a field ##\mathbb{F}## Homework EquationsThe Attempt at a Solution We can prove this entrywise. 1) Obviously the zero matrix is an upper triangular matrix, because it satisfies the...

Homework Statement Same as title. Homework EquationsThe Attempt at a Solution A defining property of a diagonal matrix is that ##A_{ij} = A_{ji} ~~\forall i,j \le n##. This means that ##((A)^{t})_{ji} = A_{ji}##. Therefore, we know that ##A^t = A##. This shows that a diagonal matrix is...

Homework Statement Not sure if this is advanced, so move it wherever. A certain rigid body may be represented by three point masses: m_1 = 1 at (1,-1,-2) m_2 = 2 at (-1,1,0) m_3 = 1 at (1,1,-2) a) find the moment of inertia tensor b) diagonalize the matrix obtaining the eigenvalues and the...

So if I have a system of equations: $$x_1+x_2+x_3=1$$ and $$x_4+x_5+x_6=1$$ Then they can be put into a matrix representation \begin{equation*} \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1\\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \\ \end{bmatrix}...

In the photo that is a matrix B. B^3 + B = 2B^2 +2I from calculations. But why not B must be equal to 2I, the reason is as follow, B(B^2 + I ) = 2(B^2 +I) B(B^2 + I)(B^2 + I)^(-1) = 2I (B^2 + I)(B^2 + I)^(-1) so B = 2I why is my concept wrong? please explain to me.

Homework Statement We're given some linear transformations, and asked what the null space, column space and row space of the matrix representations tell us Homework EquationsThe Attempt at a Solution I know what information the column space and null space contain, but what does the row space of...

Homework Statement Let A and B be n by n matrices such that A is invertible and B is not invertible. Then, AB is not invertible. Homework EquationsThe Attempt at a Solution We know that A is invertible, so there exists a matrix C such that CA = I. Then we can right -multiply by B so that CAB...

Homework Statement If A and B are square matrices of same order, prove of find a counter example that if AB = 0 then BA = 0. Homework Equations A^{-1} A = I_n, ABC = (AB)C The Attempt at a Solution AB = 0 \Rightarrow A^{-1} A B = A^{-1} 0 \Rightarrow (A^{-1} A) B = A^{-1} 0 \Rightarrow I_n...

Software for multiplication of matrices I'm going to do a lot of matrix multiplications, since I'm computing Jarlskog invariants. I would like to know if there is a great program for doing a lot of matrix multiplications? I tried with Maple but at some point it gives up. My matrices are not...

Homework Statement Hey :-) I just need some help for a short calculation. I have to show, that (\sigma \cdot a)(\sigma \cdot b) = (a \cdot b) + i \sigma \cdot (a \times b) The Attempt at a Solution I am quiet sure, that my mistake is on the right side, so I will show you my...

Homework Statement This isn't exactly a "problem" per se , but I need to understand it for a course I'm taking. I'm trying to understand the significance and when to use the vector conversion matrices, or just the identities. I'll use an example that I made up, using rectangular to polar...

I understand that the RREF algorithm can be used on matrices representing systems of equations to find the solution set of that system. However, can this algorithm be used for any matrix of any size? For example, what if we, what if we had a 1x1 mattix, or a 2x1? What is the minimum size of a...

The price of a tub of ice cream and a tub of yoghurt respectively at store A and store B are given below. Store A ( 40 , 30) Store B ( 38 , 35) (i) Represent the above information in a matrix of order 2 * 2, such that the columns denote the stores. (ii) 20 tubs of ice cream and 30...

Hello, i have here some identities for gamma matrices in n dimensions to prove and don't know how to do this. My problem is that I am not very familiar with the ⊗ in the equations. I think it should be the Kronecker-product. If someone could give me a explanation of how to work with this stuff...

Hi! I have a problem: I need to solve an equation, Ax=y, where A is a known matrix, y is a known column vector and x is an unknown column vector. Unfortunately, A is singular so I cannot do the simple solution of inverse(A)*y=x. Does anybody know of any way that I can obtain the coefficients...

Homework Statement Prove Rθ+φ =Rθ+Rφ Where Rθ is equal to the 2x2 rotational matrix [cos(θ) sin(θ), -sin(θ) cos(θ)] Homework Equations I am having a hard time trying figure our what is being asked. My question is can anyone put this into words? I am having trouble understanding what the phi...

Hey everyone, I understand how to normalize a second order system, but I wanted to know if the same steps are taken when the parameters of the system are not scalar but matrices. For example where the parameter phi, and gamma are both 3x3 matrices and X is a 3x1 vector. From what I've see...

Given a system of linear differential equations $$x_{1}'=a_{11}x_{1}+a_{12}x_{2}+\cdots a_{1n}x_{n}\\ x_{2}'=a_{21}x_{1}+a_{22}x_{2}+\cdots a_{2n}x_{n}\\ \ldots\\ x_{n}'= a_{n1}x_{1}+a_{n2}x_{2}+\cdots a_{nn}x_{n}$$ this can be rewritten in the form of a matrix equation...

Hi, I want to find the number of parameters needed to define an orthogonal transformation in Rn. As I suppose, this equals the dimension of the orthogonal group O(n,R) - but, correct me if I'm wrong. I haven't been able to figure out how to do this yet. If it helps, I know that an orthogonal...

Hello, we are given a 2×2 matrix S such that det(S)=1. I would like to find a 2x2 invertible matrix A such that: A S A^{-1} = R, where R is an orthogonal matrix. Note that the problem can be alternatively reformulated as: Is it possible to decompose a matrix S∈SL(2,ℝ) in the following way...

Homework Statement Good day, I want to ask the matrix that obtained from below formula and example. $$tr_A(L_{AB})=\sum_i [(\langle i|\otimes id)L_{AB}(|i\rangle\otimes id)]$$ this formula above can be represented as in matrix form below, $$tr_A(L_{AB})=...

Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...

A matrix is symmetric if it is equal to its own transpose, so to show $\displaystyle \begin{align*} C^T\,C \end{align*}$ is symmetric, we need to prove that $\displaystyle \begin{align*} \left( C^T\,C \right) ^T = C^T\,C \end{align*}$. $\displaystyle \begin{align*} \left( C^T\,C \right) ^T &=...

Even though school's been over since last week, I've set aside this problem because I cannot figure out how it works and the teacher only posted the solution (without the steps) on my blackboard. So, here's a system of equations that I had to solve: ##10x+24y+2z=-18## ##-2x-7y+4z=6##...

Homework Statement Let A,B be square matrices of order n. n>=2 lets A and B be matrices of Rank 1. What are the options of the Rank of A+B ? Homework EquationsThe Attempt at a Solution I know that there are 3 possibilities, 2, 1 , 0. Just having trouble with coming up with a formula. i tried...

Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?

Hey! :o I want to find all the matrices in Jordan form with characteristic polynomial the $(x+2)^2(x-5)^3$. Let $\mathcal{X} (x)=(x+2)^2(x-5)^3$. The possible minimal polynomials $m(x)$ are the ones that $m(x)\mid \mathcal{X} (x)$, so $(x+2)^2(x-5)^3$ $(x+2)(x-5)^3$ $(x+2)(x-5)^2$...

Hi, can someone please explain how we get 9 independent constants for orthotropic materials in the stiffness/compliance matrix which is reduced from 21 constants of anisotropic materials. A detailed expanation is required and mathematics behind it

Homework Statement True or False? If A is an n × n matrix, P is an n × n invertible matrix, and B = P −1AP, then a11 + a22 + . . . + ann = b11 + b22 + . . . + bn Homework Equations Diagnolization, similar matrixes The Attempt at a Solution the question is asking if the summation of the main...

Homework Statement Show that the following is NOT a vector space: {(a, 1) | a, b, c, ∈ ℝ} {(b, c) Note: this is is meant to be a 2x2 matrix. This may not have been clear in how I formatted it. 2. The attempt at a solution I am self-studying linear algebra, and have had a difficulty...

Homework Statement I need to get 7 equations to make a 7x7 matrix in order to solve it. I have 4 of them, but I'm not sure how to get the voltages equations. Homework EquationsThe Attempt at a Solution I have got four equations so far. One is simply using KCL, I1 = I2 + I3 + I4. The other...

I'll put you in context for the sake of simplicity before asking my question. Say we have the following homogeneous linear system: x'=Ax Let A be 2x2 for simplicity. Then the general solution would look like: x(t) = αa + βb And a fundamental matrix would be: Ψ(t) = ( a , b ) What...

Homework Statement Virtually all quantum mechanical calculations involving the harmonic oscillator can be done in terms of the creation and destruction operators and by satisfying the commutation relation \left[a,a^{\dagger}\right] = 1 (A) Compute the similarity transformation...

Homework Statement Hadamard matrices H0, H1, H2, . . . are defined as follows: (a) H_0 is the 1 × 1 matrix [1]. (b) For k > 0, H_k is the 2^k × 2^k matrix. \\Attached is the matrix (a) Show by induction that (H_k) ^2 = 2^k* I_k, where I_k is the identity matrix of dimension 2^k . (b) Note that...

We have ## \rho ## and a hamiltonian K on ## H_s \otimes H_E##. have we (K \rho)_S \otimes Id_E = K (\rho _S \otimes Id_E) ? here ## \rho _ s ## and ## (K \rho) _ s ## are the reduced density matrices. If P maps an operator O to ##O_S \otimes Id_E##, I have to prove that ## PK \rho = KP...

I got (very) confused about the concept of states, pure states and mixed states. Is it correct that a linear combination of pure states is another pure state? Can pure (and mixed) states only be expressed in density matrices? Is a pure state expressed in a single density matrix, whereas mixed...

Let x(t)= [x1(t) x2(t)] be a solution to the system of differential equations: x′1(t)=−2x1(t)+2x2(t) x′2(t)==−6x1(t)+9x2(t) If x(0)= [4 -2] find x(t). I got the eigenvalues to be -6 and -5, but I don't know how to calculate the coefficients in front of the exponents. For lambda=-6 I get...

Homework Statement Let A be an n x n matrix, and let v, w ∈ ℂn. Prove that Av ⋅ w = v ⋅ A†w Homework Equations † = conjugate transpose ⋅ = dot product * = conjugate T = transpose (AB)-1 = B-1A-1 (AB)-1 = BTAT (AB)* = A*B* A† = (AT)* Definitions of Unitary and Hermitian Matrices Complex Mod...

Homework Statement For each matrix A, I need to find a basis for each generalized eigenspace of ## L_A ## consisting of a union of disjoint cycles of generalized eigenvectors. Then I need to find the Jordan canonical form of A. The matrices are: ## a) \begin{pmatrix} 1 & 1\\ -1 & 3...

Hey! :o Let $M$ be a field and $G$ the multiplicative group of matrices of the form $\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}$ with $x,y,z\in M$. I want to show that $G$ is nilpotent. Could you maybe give me some hints what we could do in this case? (Wondering)...

Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...

Homework Statement I am trying to prove the following: if ##A\in C^{m\ \text{x}\ m}## is hermitian with positive definite eigenvalues, then A is positive definite. This was a fairly easy proof. The next part wants me to prove if A is positive definite, then ##\Delta_k##=\begin{bmatrix} a_{11} &...