Matrix Definition and 1000 Threads

Could some please tell me if they think my answer for 1c and 3d of these questions I've done are right. thanks.

As a step in a solution to another question our lecture notes claim that the matrix (a,b,c,d are real scalars). \begin{bmatrix} 2a & b(1+d) \\ b(1+d)& 2dc \\ \end{bmatrix} Is positive definite if the determinant is positive. Why? Since the matrix is symmetric it's positive definite if the it...

Assume the mapping T: P2 -> P2 defined by: T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 is linear.Find the matrix representation of T relative to the basis B = {1,t,t2} My book says to first compute the images of the basis vector. This is the point where I'm stuck at because I'm not...

Homework Statement I. 3*3 matrix A (8 2 -2, 2 5 4, -2 4 5) II. 3*3 matrix (1 2 0, -1 -2 0, 3 5 1) Homework Equations I. Solve Aexp 100 of 3*3 II. Find the 5th rooth of B matrix The Attempt at a Solution I. I got stuck at diagonalising the matrix. Is this OK 1st step ? If yes...

Good afternoon all, I'm investigating typical values of entropy for a subsystem of a 1D (non-interacting) spin chain. Most of the problem is essentially solved I've shown that a typical pure state of the entire chain is close (trace norm) to the state ##\Omega_S## when reduced. \Omega_S =...

So, essentially, all I wonder is: What is the The Matrix Exponent of the Identity Matrix, I? Silly question perhaps, but here follows my problem. Per definition, the Matrix Exponent of the matrix A is, e^{A} = I + A + \frac{A^2}{2} + \ldots = I + \sum_{k=1}^{\infty} \frac{A^k}{k!} =...

Hi PF Peeps! Something came up while I was studying for my QM1 class. Basically we want to represent operators as matrices and in one case the matrix element is defined by the formula : <m'|m> = \frac{h}{2\pi}\sqrt{\frac{15}{4} - m(m+1)} \delta_{m',m+1} But the thing is we know m takes on...

Homework Statement \begin{array}{rrr|r} -1 & 2 & -1 & -3 \\ 2 & 3 & α-1 & α-4 \\ 3 & 1 & α & 1 \end{array} α∈ℝ for the augmented matrix, what value of α would make the system consistent? Homework Equations N/A Answer: α=2 The Attempt at a Solution I know that the system has to have an...

The GHZ state is: |\psi> = \frac{|000> + |111>}{\sqrt2} To calculate density matrix we go from: GHZ = \frac{1}{2}(|000> + |111>)(<000| + <111|) GHZ = \frac{1}{2}( |000><000| + |111><111| + |111><000| + |000><111|) To: GHZ = 1/2[ \left( \begin{array}{cc} 1 & 0 & 0 & 0 & 0 &...

Homework Statement If matrix ## C = \left[ {\begin{array}{c} A \\ B \ \end{array} } \right]## then how is N(C), the nullspace of C, related to N(A) and N(B)? Homework Equations Ax = 0; x = N(A) The Attempt at a Solution First, I thought that the relation between A and B with C is ## C = A...

Homework Statement Given the matrices A, B, C, D, X are invertible such that (AX+BD)C=CA Find an expression for X. Homework Equations N/A Answer is A^{-1}CAC^{-1}-A^{-1}BD The Attempt at a Solution I know you can't do normal algebra for matrices. So this means A≠(AX+BD)?

Hi people. I just read some articles about physicist starting to gain more and more evidence for the Universe to be a 3D Hologram of a 2D world (or that's how I understood it). And apparently for us living in a "Matrix", like the one in the movie. Now I would like to understand the relation...

I just couldn't understand how does augmented matrix deduce inverse of a matrix. I mean what is it in the row operation because of which we get the inverse of a matrix. I just don't want to learn the steps but to understand why it works. Thank you.

Many people out there today seem to think that we'll soon have computers powerful enough to simulate the physical world well enough that we'll be able to upload ourselves and live in such a simulation. People really seem to think a Matrix situation is possible. Some, like Nick Bostrom, have...

Homework Statement I have ##A=TST(-1,2-1),## and I need to show that an eigenvector of A is,##Y_{j}=sin(kj \pi / J).## and then find the full set of eigenvalues of A. The matrix A comes from writing ##-U_{j-1}+2U-U_{j+1}=h^{2}f(x_{j}), 1\le j \le J-1##, in the form ##AU=b## Homework...

Homework Statement If A and B are 2 matrices such that AB = A and BA =B, then B2 is equal to B A Zero matrix I Homework Equations We can pre or post multiply a matrix on both sides of equation. The Attempt at a Solution (AB).(BA) = A.B AB2A = A.B Pre multiply both sides by A-1 We get B2A = AB...

Homework Statement Find an expression for e^A with the powerseries shown in the image linked Homework Equations I know you have to use eigen values and eigen vectors and a diagonal matrix The Attempt at a Solution What I did was just try to actually multiply out the infinite series given. I...

I am doing a project on image processing and I need to solve the following set of equations: nx+nz*( z(x+1,y)-z(x,y) )=0 ny+nz*( z(x+1,y)-z(x,y) )=0 and equations of the boundary (bottom and right side of the image): nx+nz*( z(x,y)-z(x-1,y) )=0 ny+nz*( z(x,y)-z(x,y-1) )=0 nx,ny,nz is the...

Question: ##h_{t}+vh_{x}+v_{x}h=0## ##v_{t}+gh_{x}+vv_{x}=0## Write it in the form ##P_{t}+Q_{x}=0##, where ##P=(h,hv)^{T}##, where ##g## is a constant ##>0##, and ##v## and ##h## are functions of ##x## and ##t##. Attempt: I have ##Q=(vh,?)^{T}##, the first equation looks easy enough, but...

Homework Statement Find the inverse of the matrix: 1 1 -1 2 -1 1 1 1 2 Homework Equations One must be aware of the identity matrix, as well as how add one row to another with matrix multiplication, for example, the matrix 1 0 0 k 1 0 0 0 1 would add k times the first row to the second...

I'm working through Meisner Thorne and Wheeler (MTW), but have been temporarily sidetracked by a problem with rotation matrices. I've worked through the maths and produced the matrices by multiplying the three individual rotation matrices, (no problem there) but I have been trying to work out...

Homework Statement I have 4 equations. 3x+6y-5z-t=-8 6x-2y+3z+2t=13 4x-3y-z-3t=-1 5x+6y-3z+4t=-6 I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2 Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors...

Homework Statement Here's my problem. I only need help with the bottom part, but if you could explain the problem more vividly that would help too. Homework Equations A = S-1BS (?) There aren't really any relevant equations. This part of linear algebra is getting really abstract, at least I...

In a given matrix A, the singular value decomposition (SVD), yields A=USV`. Now let's make dimension reduction of the matrix by keeping only one column vector from U, one singular value from S and one row vector from V`. Then do another SVD of the resulted rank reduced matrix Ar. Now, if Ar is...

Hello I want to obtain ABCD matrix element value.At first I tried to find A element value with boundary conditions but I don't know how can I find relationship between V(1)+ and V(1)- . Any help appreciate

Homework Statement If A is a square matrix of order 3 then the true statement is 1. det(-A) = - det A 2.det A = 0 3.det ( A + I) = I + detA 4.det(2A) = 2detA Homework Equations NA The Attempt at a Solution 2. option is obviously not true. Making a random matrix A and verifying properties 1. ...

True or False: If the linear combination x -> Ax maps Rn into Rn, then the row reduced echelon form of A is I. I don't understand why this is False. My book says it is false because it is only true if it maps Rn ONTO Rn instead of Rn INTO Rn. What difference does the word into make?

Can someone confirm or refute my thinking regarding the diagonalizability of an orthogonal matrix and whether it's symmetrical? A = [b1, b2, ..., bn] | H = Span {b1, b2, ..., bn}. Based on the definition of the span, we can conclude that all of vectors within A are linearly independent...

Getting stuck on something I think that could be trivial. Maybe someone can see my mistake. consider the system: $x' = -2x + y$ and $y' = 2x - 3y$ a) Write the system in matrix form my solution $\overrightarrow{X} = (^x_y)$ so: $X' = (^{x'}_{y'})$ so $A = $ \begin{bmatrix} -2 & 1 \\ 2 & -3...

Homework Statement Not a homework problem. Typically, we are given a matrix, then asked to find the basis for the kernel or image space of the matrix. I've never seen a problem that did the converse (i.e., given the matrix for the kernel/image space of some matrix, find some matrix). I was...

So, B-L is a U(1) generator extracted out of some unified theories of leptons and quarks and in such theories it is traceless, with B=1/3 and L=1, and the trace taken over a "four coloured" multiplet, namely a lepton and three colored quarks. Now, I am amazed that there is another Matrix that...

For the distance function ##(\Delta s)^2 = (\Delta r)^2 + (r \Delta \theta)^2##, the rotation matrix is ##R(\theta) = \begin{pmatrix} cos\ \theta & - sin\ \theta \\ sin\ \theta & cos\ \theta \end{pmatrix}##. That means that for the distance function ##(\Delta s)^2 = (\Delta r)^2 +...

Seems exist some relationship between the inverse of a matrix with the inertia tensor, looks: This relationship really exist?

Simple question really, I'm not sure why the constant pulled out of the derivative becomes negative (-w2). I've tried looking for answers by googling but can't come up with anything. I feel like its because the first term (1,1) is negative but I want to be sure. Thanks

Consider a function ##f : U \subseteq \mathbb{R}^{n} -> \mathbb{R}## that is an element of ##C^{2}## which has an minimum in ##p \in U##. According to Taylor's theorem for multiple variable functions, for each ##h \in U## there exists a ##t \in ]0,1[## such that : ##f(p+h)-f(p) =...

Hello! (Wave) Suppose that we are given a directed graph and we want to find out if a vertex $j$ is reachable from another vertex $i$ for all vertex pairs $(i, j)$ in the given graph. Reachable mean that there is a path from vertex $i$ to $j$. The reachability matrix is called transitive...

Homework Statement Consider a symmetric n x n matrix ##A## with ##A^2=A##. Is the linear transformation ##T(\vec{x})=A\vec{x}## necessarily the orthogonal projection onto a subspace of ##R^n##? Homework Equations Symmetric matrix means ##A=A^T## An orthogonal projection matrix is given by...

Homework Statement I was working through my text on deriving the tensor for Angular momentum of the sums of elements of a rigid body, I follow it all except for one step. Here is a great page which shows the derivation nicely - http://www.kwon3d.com/theory/moi/iten.html I follow clearly to the...

Homework Statement Not really a homework question. Something that I've been wondering about. The distributive law holds for matrices. Let A and B be n x n matrices. Why is the following true for all A&B? ##(A+B)^2=A^2+2AB+B^2## I don't undrestand that middle term (2AB) and why there's a...

Homework Statement Claim: If ##A \in \mathcal{M}_n (\mathbb{C})## is arbitrary, and ##D## is a matrix with ##\beta## in its ##(i-j)##-th entry, and ##\overline{\beta}## in its ##(j-i)##-th, where ##i \ne j##, and with zeros elsewhere, then ##Tr(AD) = a_{ij} \beta + a_{ji} \overline{\beta}##...

Be the matrix A defined by [aijk] (a matrix 2x2x2), do you know how to compute a inverse this cubic matrix?

I'm sure that this problem is easier than I am making out to be, but I'm going over some review problems for an exit exam and I'm having a little trouble with this one. Let the matrix $A$ be given by: $$A = \begin{pmatrix} 1&4\\ 2&3 \end{pmatrix}$$ Find $A^n$ for general $n$. I have the...

Hi, I want to show that the Trace of the Product of a symetric Matrix (say A) and an antisymetric (B) Matrix is zero. $So\: (AB)_{ij}=\sum_{k}^{}{a}_{ik}{b}_{kj} $ $and\: Tr(AB)=\sum_{i=j}^{}(AB)_{ij}=\sum_{i}^{}\sum_{k}^{}{a}_{ik}{b}_{ki} $ $because\:A\:is\:symetric, \: {a}_{ik}=...

Homework Statement Let T : R2→R2, be the matrix operator for reflection across the line L : y = -x a. Find the standard matrix [T] by finding T(e1) and T(e2) b. Find a non-zero vector x such that T(x) = x c. Find a vector in the domain of T for which T(x,y) = (-3,5) Homework EquationsThe...

Homework Statement The Matrix P = 1 0 3 1 1 0 0 3 1 is the transition matrix from what basis B to the basis B' = {(1,0,0),(1,1,0),(1,1,1) for R3? Homework Equations [v]B=P[v]B' The Attempt at a Solution I'm looking...

Homework Statement Please see the attached file if my inline insertion does not work. Homework Equations ##det(A)=det(A^T)##[/B]The Attempt at a Solution Since a matrix has a determinant of zero only when it's columns are linearly dependent, we look for a set of points [x1 x2] such that...

Suppose a position vector v is rotated anticlockwise at an angle ##\theta## about an arbitrary axis pointing in the direction of a position vector p, what is the rotation matrix R such that Rv gives the position vector after the rotation? Suppose p = ##\begin{pmatrix}1\\1\\1\end{pmatrix}## and...

I have my state vector containing $$[X, Y, v_x, v_y, \theta, r, a_x, a_y, b_{\theta}]^T$$ and I have them related by $$dX = v_x cos \theta - v_y sin \theta\\ dY = v_x sin \theta + v_y cos \theta\\ dv_x = a_x\\ dv_y = a_y\\ d\theta = r\\ dr = 0\\ da_x = 0\\ da_y = 0\\ db_\theta = 0\\ $$ Now...

When you diagonalize a matrix the diagonal elements are the eigenvalues but how do you know which order to put the eigenvalues in the diagonal elements as different orders give different matrices ? Thanks

I would like to understand how LCAO may be used to construct the matrix to be solved for the molecular orbitals of two cases of molecules: 1) small molecules like H3 (or H3+, HF2-, H2O, CH4, etc. 2) groups or parts of molecules with delocalized pi-systems (including linear and cyclic...