Matrices Definition and 1000 Threads

What is the purpose and meaning of Pauli/sigma matrices ? What distinguish them from other Operators ? thanx

Hi! I can define \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3 I know that the four gamma matrices \gamma^i\:\:,\;i=0...3 are invariant under a Lorentz transformation. So I can say that also \gamma ^5 is invariant, because it is a product of invariant matrices. But this equality holds: \gamma...

Matrices- Variable in Matrices-Help! 1-19-12 I need help. I have a homework question that I have tried to solve. The matrix in the attachment has no inverse. Explain how you can determine the value of x. Then find x. I tried cross multiplying. 3*2/3=4x 6/3=4x 2=4x x=2/4 x=1/2

I'm just revising my maths notes on matrices and I have a couple of questions, mainly about representation of matrices. 1. I have drawn the matrices in my notes with square brackets, curved brackets or straight lines and curved ends. I always just copied what the slide show or board notes...

I'm just wondering how the Majorana matrices first were found. I have only seen them immediately written down at different webpages, and never seen a derivation. Is it obvious how to transform the Dirac gamma matrices into the Majorana representation?

hi all, Is that skyline storage,which is been widely used in FEM problems, is only for symetric matrices? What if I have non-standardized matrices, that is , which can not be made symettric, has pretty randomly oriented inner products, which can not be put into any computerized manner for...

Often people asks how to obtain a positive definite matrix. I would like to make a list of all possible ways to generate positive definite matrices (I consider only square real matrices here). Please help me to complete it. Here M is any matrix, P any positive definite matrix and D any...

Homework Statement Find all 10x10 Matrices such that the column space is equal to the null space. Homework Equations Choose Function: n!/k!(n-k)! where n is the total number of elements and k is the number k-cominations of the set. rankA+dimNulA=n for a matrix in R^n The...

Can someone point me to the proof or give it here for the claim that product of the two positive definite (real) matrices is positive definite. How about determinants of two matrices? Is det(AB) = det(BA) Rank(AB) = Rank(A)Rank(B)? Thank you in advance. This is not a homework...

Homework Statement Consider a 4 x 4 matrix A = B C 0 D where B, C, and D are 2 x 2 matrices. What is the relationship between the eigenvalues of A, B, C, and D? The Attempt at a Solution I suppose you can write A as: b1 b2 c1 c2 b3 b4 c3 c4 0 0 d1 d2 0 0 d3...

Homework Statement "Let T:M2,3→M3,2 be represented by T(A) = AT. Find the matrix for T relative to the standard bases for M2,3 and M3,2"Homework Equations I let the transformation matrix be B. I know that BA = AT, so I need some matrix times A to equal A transpose.The Attempt at a Solution I'm...

Homework Statement Suppose matrices A and B are similar. Explain why they have the same rank. Homework Equations The Attempt at a Solution So if A and B are similar, then there is some invertible matrix P such that B = P^-1AP. I have been trying to find some way to relate...

Homework Statement there are 13,200 components that must be regularly maintained or else they fail. At the end of month “t”, there are is a state vector describing the components given by Vt = [ft, mt, at]T where ft is the number of failed components, mt is the number of components out...

Homework Statement If the rows of A are linearly dependent, prove that the rows of AB are also linearly dependent.The Attempt at a Solution A = \begin{pmatrix}a&-a\\b&-b\end{pmatrix} the rows are linearly dependent because a - a = 0 and b - b = 0. B =...

Hey guys, I was wondering how to get the expression for pauli matrices. I know that for one electron: S_i = \frac{\hbar}{2} \sigma_i But I also know that you can get to the above expression by explicitly calculating the matrix elements of the Sz, Sx and Sy operators (in the basis generated...

Hi, I work in NRM and need for some reason to optimize an objective function of the form ||M-M_target||^2 where M is the product of a large number (>100) 2D unitary complex matrices (Qi) and a vector (A), i.e. M=Q1*Q2*...*QN*A, and M_target is a constant complex vector. I can do it directly...

Homework Statement Prove I=T1+T2+...+Tk Where Ti=pi(T) Homework Equations T is kxk pi(x)=(x-c1)...(x-ck) is the minimal polynomial of T. pi=\pii(x)/\pii(ci) \pii=\pi(x)/(x-ci) To evaluate these functions at a matrix, simply let ci=ciI The Attempt at a Solution From lagrange interpolation...

An augmented matrix scaled by a number also means the solutions set is scaled by that same number. I believe this is true due to it basically being the same as elementary row operations preformed on each row. Unless it is a zero scalar in which case you lose all conditions. Is my method of...

Hi guys, I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars) However, then I was asked...

Homework Statement Find the values of a and b such that the equations: 3x + ay = 2 and -6x + 4y = b have i) an infinite set of solutions ii) no solutions The Attempt at a Solution \begin{pmatrix} 3 & a \\ -6 & 4 \end{pmatrix} * \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} 2\\ b...

If A and B are both invertible square matrices of the same size with complex entries, there exists a complex scalar c such that A+cB is noninvertible. I know this to be true, but I can't prove it. I tried working with determinants, but a specific selection of c can only get rid of one entry...

I urgently need some help in my problem for my MS thesis. I have two datasets of same variable dimension but different number of observations, ie same # of columns but not same # rows. The variables are indentical for both sets. I want to compare the multivariate distributions of the two data...

I have an (m \times n) complex matrix, \textbf{N}, whose elements are zero-mean random variables. I have a sort of covariance expression: \mathcal{E}\left\{\textbf{N}\textbf{N}^H\right\} = \textbf{I} where \mathcal{E}\left\{\right\} denotes expectation, \{\}^H is conjugate transpose and...

Homework Statement Find a third column so that U is unitary. How much freedom in column 3? [ 1/√3 i/√2 ] [1/√3 0 ] = U [i/√3 1/√2 ] Homework Equations UHU=I UH=U-1 The Attempt at a Solution Obviously in order for the matrix to be unitary the...

Homework Statement What can you say about a. the sum of a complex number and its conjugate? b. the conjugate of anumber on the unit circle? c. the product of two numbers on the unit circle? d. the sum of two numbers on the unit circle? Homework Equations The Attempt at a...

find B matrices so B^{3}=A=\left(\begin{array}{cc}14 & 13\\13 & 14\end{array}\right) ,the diagonal form of A is D=\left(\begin{array}{cc}a & 0\\0 & b\end{array}\right) i got weird numbers so for convinience the eigenvalues are a,b so there is U for which...

Homework Statement H is a nxn matrix with elements in {0,1} G is a nxn matrix with elements in GF(2) m is a nx1 vector with elements in GF(2). How can we perceive the output of HGm where Gm multiplication is in GF(2) and H multiplication is a normal real multiplication. Actually I want...

Homework Statement Suppose V is a unitary space [over C] and T: V -> V is a normal transformation that satisfies T-1=-T. Prove that T is unitary transformation. Homework Equations I know that T is unitary if and only if it is normal and the absolute value of its eigenvalues is 1. [*2] The...

Homework Statement I don't understand why the Pauli matrix σx is hermitian. Nonetheless, I am able to prove why the σy matrix is hermitian. Homework Equations The Attempt at a Solution Whenever I do the transpose and then the conjugate I get the negative of σx instead. Am I doing...

Homework Statement Show that the matrices \mathcal{D}_{m',m}^{(j)}=\langle j,m'|\exp(-\frac{i}{\hbar}\vec{J}\hat{n}\Phi)|j,m\rangle form a group (i.e. multiplication, inverse and identity). No idea how to even begin.

Homework Statement Alvin is informed that the homogeneous system of equations AX = 0 has a one parameter family of solutions given by X = t[4 3 0]T By trial and error, he has found that X = [-1 5 7]T is a solution to the inhomogenous problem AX = B where B = [3 -2 -5]T...

How does knowing that two matrices anticommute AB=-BA and that A^2=1 and B^2=1 help me to know how to find the trace of the matrices. I am supposed to show that their traces equal each other which equals 0 but I am not sure exactly how the given information helps me determine the trace?

Does this have to do with the eigenvalues?

Hello! I'm trying to write an essay on RQM. The problem I have encountered is the diffrent choices of matrices for the dirac equation. The two choices that I´m mixing up in my equations are: \begin{eqnarray} \gamma^0 = \left( \begin{array}{cc} I & 0 \\ 0 & -I \end{array} \right), \quad...

Given any two n X n matrices A and B , prove or disprove the following statements: If AB=-BA then at least one of A and B is singular.

Homework Statement We are looking for the matrix A Homework Equations (A^transpose)^transpose=A The Attempt at a Solution i would start with finding the transpose of the matrix. -5 0 -8 -7

A graph can be represented by an adjacency matrix but how is that a real mathematical matrix and not just a table? A matrix is part of an equation system Ax=B but what is x and B in this case if A is the adjacency matrix? For example Google does PageRank with Eigenvalues but what would...

Homework Statement The costs (in millions of dollars) of connecting any two of the four cities A,B,C and D by telephone lines are given in the following matrix: 0 3 5 4 3 0 2 3 5 2 0 6 4 3 6 0 a) Draw a diagram of the complete graph b) find a minimal spanning tree The...

I've just started a masters in physics after a 4-year break and am having some real trouble getting back into the swing of things! We have been asked to prove some properties of gamma matrices, namely: 1. \gamma^{\mu+}=\gamma^{0}\gamma^{\mu}\gamma^{0} 2. that the matrices have eigenvalues...

Is any symmetric matrix diagonalisable with an orthogonal change of basis? Does the minimal polynomial of any real matrix split into distinct linear factors? Is a real inner product an example of a bilinear form? Could 2 complex matrices which are similar have different Jordan normal forms?

In post #7 of https://www.physicsforums.com/showthread.php?t=532666" thread, the OP asked whether one could meaningfully divide by a matrix. Certainly this is possible for invertible matrices, but I'm wondering if it's possible to define something similar even for singular matrices. For...

My textbook is using Lagrange multipliers in a way I'm not familiar with. F(w,λ)=wCwT-λ(wuT-1) Why is the first order necessary condition?: 2wC-λu=0 Is it because: \nablaF=2wC-λu Why does \nablaF equal this? Many thanks! Edit: C is a covariance matrix

Homework Statement 28. Let A be an m x n matrix with a row consisting entirely of zeros. Show that if B is an n x p matrix, then BA has a row of zeros. Homework Equations N/AThe Attempt at a Solution A = (aij)_{mxn} and B = (bij)_{nxp}. Assuming that the entries for jth column of A are all...

Homework Statement Hi A matrix M has an inverse iff it is of full column and row rank, and row rank = column rank. Since any orthogonal matrix has full column rank, does that imply that non-singular matrices are orthogonal as well? Cheers, Niles.

I was wondering if there is a representation of gamma matrices unitarily equivalent to the standard representation for which Dirac Spinors with positiv energy and generic momentum have only the first two component different prom zero. Anyone can help me?

Why are orthogonal matrices important?

Homework Statement Given p(x) = x4+2x2+1 and A = [[1 1 -2 0] [0 1 0 2] [1 1 -1 1] [0 0 -2 -1]] p(A) = 0 Find a polynomial q(x) so that q(A) = A-1 a) What is q(x)? b) Compute q(A) = A-1 Homework Equations I found the Cayley-Hamilton theorem, which states: p(x) = det(A-xIn)...

1) I can assume all these matrices to be 2x2. We have matrix A and B and AB = BA, that is, they commute.. Prove if C = A^2 + 2*A and D = A^3 + 5 * I (I is identity matrix), then CD = DC. Then give a theory that generalizes this. 2) why does R(theta)R(phi)=R(theta+phi)? (explain with...

Hi, The typical representation of the Dirac gamma matrices are designed for the +--- metric. For example /gamma^0 = [1 & 0 \\ 0 & -1] , /gamma^i = [0 & /sigma^i \\ - /sigma^i & 0] this corresponds to the metric +--- Does anyone know a representation of the gamma matrices for -+++...

The lecturer said that a way to find the determinant of a matrix is to do the following det(A) = xdet(B) (1) where A is the original matrix, B is an arbirtray matrix and x is a scalar multiplier The lecturer also said that a simple way to find the determinant of a high...