Matrix Definition and 1000 Threads

The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...

What is the basic matrix form for a uniform (unit) sphere centered at the origin? Given a vector that specifies the radii (1,1,1) == (r1,r2,r3), I would like the matrix that implies no rotation (is it [[1,0,0],[0,1,0],[0,0,1]]?) and covers the rest of the necessary parameters. I am testing...

Given the Lorentz matrix Λuv its transpose is Λvu but what is its transpose ? I have seen ΛuaΛub = δb a which implies an inverse. This seems to be some sort of swapping rows and columns but to get the inverse you also need to replace v with -v ? Also In the LT matrix is it the 1st slot...

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...

Homework Statement Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)} a) Show that B1 is a basis for [R][/3] b) Find the coordinates of w=(2,3,1) relative to B1 c)Given that B2 is a basis for [R[/3], find...

We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis. The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or...

Homework Statement A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...

Homework Statement Below are four equations, with the known quantities listed. Solve these equations to obtain an expression for ##T## in terms of known quantities only. Do the same to obtain an expression for ##a## ##T-f=m_1a\hspace{5mm}N-m_1g\cos\theta=0## ##m_2g-T=m_2a \hspace{5mm} f=\mu N##...

Homework Statement Okay I am given a matrix A = [2 1 ; 3 4] The first step is to find numbers of a and b such that A2 + aA + bI = [0 0; 0 0] I is an identity matrix (2x2). Part B - After that is says to use the result of the above to express A5 as a linear combination of A and I Homework...

Hello everyone, I'm struggling with a coupled of matrix equations of the general form: AX + CY = cX BY + DX = cY where A, B, C and D are hermitics square matrices. X, Y and c are the eigenvector and eigenvalue to be found. I'm looking for a method or an algorithm to solve this system by using...

I am creating a gray-scale image of a 2000*2000 matrix using mat2gray and imshow command.But highest number of matrix entries that imshow can implement is 500*500 approximately.After that it shows------ "Warning: Image is too big to fit on screen; displaying at 8% > In...

Hey JO. The Hamiltonian is: H= \frac{p_{x}^{2}+p_{y}^{2}}{2m} In quantum Mechanics: \hat{H}=-\frac{\hbar^{2}}{2m}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial x^{2}}) In polar coordinates: \hat{H}=-\frac{\hbar^{2}}{2m}( \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}...

Homework Statement Say you've been given vectors v1, v2 and v3. Homework EquationsThe Attempt at a Solution How do I construct a matrix out of these three vectors? Am I to use the given vectors as columns or rows in a matrix? When does this matter and when does it not? This may be a stupid...

1. Get A from its inverse3. I believe it has something to do with the theorem that states: E1E2E3...EkA=I

Homework Statement [/B] Find a 2 x 2 matrix that maps e1 to –e2 and e2 to e1+3e2Homework Equations [/B] See the above notesThe Attempt at a Solution [/B] I am making a pig's ear out of this one. I think I can get e1 to –e2 3 -1 1 -3 but as far as getting it to reconcile a matrix like...

Under some circumstances, whenever I call DEVCCG to diagonalize a general complex matrix, the program gets stuck inside and never returns. I do not even get out an error code so that I may continue with the rest of the program. I assume the iterative diagonalization inside the procedure does not...

I have questions regarding the 24 gauge bosons of the SU(5) model. I keep seeing this matrix popping up in the documents I'm reading with no real explanation of why: First of all I'm wondering how this is constructed, which means I'm wondering what the V_{\mu}^{a} look like (I already have...

In my school LA requires a pre req proof class vs matrix algebra which doesn't. Would matrix algebra even be worth taking?

Hi all, Firstly, I am not sure whether this is the area of the forum to ask this. I have been learning and researching a completely different topic, and from this I have come across a completely new concept of the Kronecker function. I have done a google search on this to get the intro and...

Homework Statement Find a 2X2 matrix that has all non-zero entries where 3 is an eigenvalue Homework EquationsThe Attempt at a Solution well since the 2x2 matrix cannot be triangular, it makes things harder for me. I have no idea where to start. I am not given any eigenvectors either. It seems...

Homework Statement A square matrix ##n\times n##, A, that isn't the zero-matrix have powers ##A^{k-1}## that isn't the zero matrix. ##A^k## is the zero matrix. What are the possible values for ##k##? Homework Equations N/A The Attempt at a Solution I'm a bit lost here but I figure that maybe...

Homework Statement Sakurai Modern Quantum Mechanics Revised Edition. Page 81. density matrix p = 3/4 [1 0; 0 0] + 1/4 [1/2 1/2; 1/2 1/2]. We leave it as an exercise to the reader the task of showing this ensemble can be decomposed in ways other than 3.4.24Homework Equations 3.4.24 w( sz +...

Homework Statement Show that strictly upper triangular ##n\times n## matrices are nilpotent. Homework EquationsThe Attempt at a Solution Let ##f## be the endomorphism represented by the strict upper triangular matrix ##M## in basis ##{\cal B} = (e_1,...,e_n)##. We have that ##f(e_k) \in...

Homework Statement Let ##U## be a ##2\times 2## orthogonal matrix with ##\det U = 1##. Prove that ##U## is a rotation matrix. Homework EquationsThe Attempt at a Solution Well, my strategy was to simply write the matrix as $$U = \begin{pmatrix} a & b\\ c & d \end{pmatrix}$$ and using the given...

I am reading a paper and am stuck on the following snippet. Given two orthonormal frames of vectors ##(\bf n1,n2,n3)## and ##(\bf n'1,n'2,n'3)## we can form two matrices ##N= (\bf n1,n2,n3)## and ##N' =(\bf n'1,n'2,n'3)##. In the case of a rigid body, where the two frames are related via...

This page (https://shiyuzhao.wordpress.com/2011/06/08/rotation-matrix-angle-axis-angular-velocity/), gives the following relation: \left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T} Where: * ##R## is a DCM (Direction Cosine Matrix) * ##\vec{v}## is the angular...

Do you know any books or reviews that explains these in sufficient detail? I am having some small problems in understanding the triangles of the CKM matrix elements and experiments conducted for their measurement...

How many ways to arrange cells of k possible values in a mxn matrix provided that sums of all rows and columns are known? For example, if we have a 5x3 matrix and 10 possible values ( from 0 to 9) that can be assigned for each cell, then how many ways to arrange cells in that matrix satisfying...

Hello. I'm having trouble understanding what is required in the following problem: Find the relation between the matrix elements of the operators $\widehat{p}$ and $\widehat{x}$ in the base of eigenvectors of the Hamiltonian for one particle, that is, $$\widehat{H} = \frac{1}{2M} \widehat{p}^2...

Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu

Hi Folks, I am looking at Shankars Principles of Quantum Mechanics. For Hermitian Matrices M^1, M^2, M^3, M^4 that obey M^iM^j+M^jM^i=2 \delta^{ij}I, i,j=1...4 Show that eigenvalues of M^i are \pm1 Hint: Go to eigenbasis of M^i and use equation i=j. Not sure how to start this? Should I...

Hi Folks, I calculate the eigenvalues of \begin{bmatrix}\cos \theta& \sin \theta \\ - \sin \theta & \cos \theta \end{bmatrix} to be \lambda_1=e^{i \theta} and \lambda_2=e^{-i \theta} for \lambda_1=e^{i \theta}=\cos \theta + i \sin \theta I calculate the eigenvector via A \lambda = \lambda V as...

I've seen various different matrices used to represent beam splitters, and am wondering which is the "right" one. Alternatively, are there various kinds of beam splitters but everyone just ambiguously calls them the same thing? The matrices I've seen are the...

Homework Statement Find a non zero matrix(3x3) that does not have in its range. Make sure your matrix does as it should.The Attempt at a Solution [/B] I know a range is a set of output vectors, Can anyone help me clarify the question? I'm just not sure specifically what its asking of me, in...

I calculate 1) ##\Omega= \begin{bmatrix} 1 & 3 &1 \\ 0 & 2 &0 \\ 0& 1 & 4 \end{bmatrix}## as not Hermtian since ##\Omega\ne\Omega^{\dagger}## where##\Omega^{\dagger}=(\Omega^T)^*## 2) ##\Omega\Omega^{T}\ne I## implies eigenvectors are not orthogonal. Is this correct?

Homework Statement [/B]Find the matrix that performs the operation 2x2 Matrix which sends e1→e2 and e2→e1Homework EquationsThe Attempt at a Solution [/B] I know e1 = < 1 , 0> and e2 = <0 , 1> Basically I'm not quite sure what the question is asking. This is the one of the problems I am...

Let A be a 2 X 2 matrix such that AX = XA for all 2 X 2 real matrices X. Show that A =kI for some k belonging to R

A difference matrix takes the entries of a vector and computes the differences between the entries like [x1 - 0 ] = difference from 0 and x1: 1 step [x2 - x1] = difference from x2 and x1: 1 step [x3 - x2] = difference from x3 and x2: 1 step assuming we had a vector x in Ax = b So why now when...

It's a new series on Netflix from those involved with Babylon 5 and the Matrix. Has anyone here seen it yet? What did you think about it?

Hi everyone I am trying to diagonalise a (2n+1)x(2n+1) matrix which has diagonal terms A_ll = (-n+l)^2 and other non vanishing terms are A_l(l+1) = A_(l+1)l = constant. Is there any way I can solve it for general n without having to use any numerical methods. I remember once a professor...

If I have two random variables X, Y that are given from the following formula: X= \mu_x \big(1 + G_1(0, \sigma_1) + G_2(0, \sigma_2) \big) Y= \mu_y \big(1 + G_3(0, \sigma_1) + G_2(0, \sigma_2) \big) Where G(\mu, \sigma) are gaussians with mean \mu=0 here and std some number. How can I find...

say for example when I calculate an eigenvector for a particular eigenvalue and get something like \begin{bmatrix} 1\\ \frac{1}{3} \end{bmatrix} but the answers on the book are \begin{bmatrix} 3\\ 1 \end{bmatrix} Would my answers still be considered correct?

Homework Statement Let A be the matrix \left(\begin{array}{cc}a&b\\c&d\end{array}\right), where no one of a, b, c, d is zero. It is required to find the non-zero 2x2 matrix X such that AX + XA = 0, where 0 is the zero 2x2 matrix. Prove that either (a) a + d = 0, in which case the general...

Hello everybody, Sorry to ask you something that may be easy for you but I'm stuck. For example I have 2 images (size 2056x2056). One image of reference and the other is the same rotated from -90degrees. Using a program with keypoints, it gives me a transform matrix : a=2.056884522e+03...

I have the following matrix R(x) = [cos(x) -sin(x) ; sin(x) cos(x)] Now consider the unit vectors v = [1;0] and w = [0,1]. Now if we compute R(x)v and R(x)w the vectors are supposed to rotate about the origin by the angle x in a counter clockwise direction. I am struggling to see how this...

Ok so officially a matrix is a rectangular array of numbers, symbols, etc arranged in rows and columns that is treated in certain prescribed ways. But that doesn't help me understand a darn thing. From what I understand, a matrix is a math tool that can help you solve linear systems, represent...

Homework Statement _ _ |1 0 0 0 -1 0 0 | 700 | |0 1 0 0 -1 0 0 | 500 | |0 0 1 0 0 0 0 0 | 150 | |0 0 0 1 0 1 0 | 1200 | |0 0 0 0 1 0 0 | -650 | |0 0 0 0 0 0 1 | -600 | Homework EquationsThe Attempt at a Solution This is driving me up the wall, am I missing...

Consider I+A*B where A: (n*l) is a variable matrix and B: (l*n) is known. I am looking for some way to find a sufficient condition for nonsingularity of I+A*B

Hi Since a few days I've been confused about the seesaw mass matrix explaining neutrino masses. It is the following matrix: \begin{pmatrix} 0 & m\\ m & M \\ \end{pmatrix}. As can easily be checked it has two eigenvalues which are given by M and -m^2/M in the limit M>>m (the limit doesn't...

Hello Everyone, Can someone explain why do matrices with linearly independent columns have only 0 vector in their null space? Thanks