Matrices Definition and 1000 Threads

I am studying the subject of linear dependence right now and had a question on this topic. Is it possible to construct a square matrix A such that the columns of A are linearly dependent, but the columns of the transpose of A are linearly independent? My intuition tells me no, but I'm not sure...

Hi all, I'd like to define a vector valued function in mathematica 7 as the exponential of a quadratic form, defined with respect to a purely symbolic matrix. What I want to do with it is to take derivatives with respect to the components of my vector, and evaluate the result when all...

1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O. Can you show me the steps of solving this problem? Please!

I am curious how to derive the (I+N)^-1 = I - N + N^2 - N^3 + ... N^(k-1) + 0 Where N^k = O, because we assume that N is nilpotent. Actually I'm just supposed to show that the inverse always exists (for my homework), but I'm not asking how to find existence, I want to know how this equation...

Can anyone explain to me how to count the total # of non-invertible 2x2 matrices? I have the answer from the book, which is r^3+r^2-r provided r is a prime. But it doesn't explain how to get there, and I couldn't figure it out. I haven't been practicing linear algebra for quite a long...

The problem: I need to find the (minimal) rank of some matrix which is basically all parameters. For example, when i ask for the rank of \begin{pmatrix} a& b& c \\ d& e& f \\ g& h& i \end{pmatrix}, I get 3. I would like to get 1, since (excluding the possibility of a matrix of all 0's) it...

Homework Statement Consider a linear system of four equations with three unknowns. We are told that the system has a unique solution. What does the rref of the co efficient matrix look like? Homework Equations The Attempt at a Solution When it says "unique solution" I'm going to...

Homework Statement Find all solutions using Gauss-Jordan elimination: [ 0 0 0 1 2 -1 l 2 1 2 0 0 1 -1 l 0 1 2 2 0 -1 1 l 2] Homework Equations Switching rows, able to scale any row able to add non zero multiple to row The Attempt at a Solution What I did was...

Homework Statement A school has three clubs, and each student is required to belong to exactly one club. One year the students switched club membership as follows: Club A. 1/5 remain in A, 2/5 switch to B, and 2/5 switch to C. Club B. 1/4 remain in B, 1/2 switch to A, and 1/4 switch to...

You know the example "The space of functions from a set S to a field F" that's usually given in a linear algebra text? Well they never give an example of the set they're working in in detail so I defined the set as: ((S, (S x S, S, +)), ((F, (F x F, F, +')), (F x F, F, °)), (S x F, F, •))...

I am currently reading Dirac Equation from Peskin-Schroeder. In a particular para it says, "Now let us find Dirac Matrices \gamma^\mu for four-dimensional Minkowski Space. It turns out that these matrices must be at least 4X4." What is the proof of the above statement? I think (not sure)...

1) show that if AB = AC and A is nonsingular, then B = C. 2) show that if A is nonsingular and AB = 0 for an n x n matrix B, then B = 0. 3) Consider the homogenous system Ax=0, where A is n x n. If A is nonsingular, show that the only solution is the trivial one, x=0. 4) Prove that if A...

Say I was given a 2x2 matrix made from a certain basis {|x\rangle, |y\rangle} , and I split that matrix into two parts, one being the diagonal part and one being the off-diagonal part. for example, if I had H = H_0 + W = \left(\begin{array}{cc}a&c\\b&d\end{array}\right) =...

Simultaneous "eigenspace" of non-commuting matrices Hello! I have been working on the following "brain teaser" the whole day long without any success. I am not even sure there is a "clean" solution. I would love to hear your opinion. Before presenting the whole problem, here is an easy...

Let C1 and C2 be circles in the plane. Describe the number of possible points of intersection of C1 and C2. It is 4?

1) Is the matrix [upper row 3 0 and lower row 0 2] a linear combination of the matrices [upper row 1 0 and lower row 0 1] and [upper row 1 0 and lower row 0 0]? Justify your answer. Is it I just have to add the two matrices to see if they are equal the matrix, [upper row 3 0 and lower row 0...

[b]1. Let M(2,R) be the set of all 2 x 2 matrics over R. Give an example of matrices A,B,C in M(2,R) such that AB=AC, but B is not equal to C. [b]3.

solve each given linear system by the method of elimination 2x-3y+4z=-12 x-2y+z=-5 3x+y+2z=1 How to solve this problem. What should I do first?

Can anyone prove that the eigenvectors of symmetric matrices are orthogonal?

Not a Homework problem, but I think it belongs here. Homework Statement Consider four dirac matrices that obey M_i M_j + M_j M_i = 2 \delta_{ij} I knowing the property that Tr ABC = Tr CAB = Tr BCA show that the matrices are traceless. Homework Equations Tr MN = Tr NM The Attempt...

[b]1. Homework Statement [/ from the ets general physics practice test (ill take it in april) the state of spin 1/2 particles using the eigenstates up and down Sz up= 1/2 hbar Sz down= -1/2 hbar Homework Equations given sigmax (pauli spin matrix) which of the following list...

In most of the physics textbooks I read they only give one or two representations of gamma matrices, but none gives a proof, so how can I prove it from the Clifford algebra?

Homework Statement Let A, B, C, D be nxn complex matrices such that AB and CD are Hermitian, i.e., (AB)*=AB and (CD)*=CD. Show that AD-B*C*=I implies that DA-BC=I The symbol * indicates the conjugate transpose of a matrix, i.e., M* is the conjugate transpose of M. I refers to the identity...

Can a symmetric matrix contain complex elements(terms). If no, how is it that 'eigen values of a symmetric matrix are always real'(from a theorem) Is a symmetric matrix containing complex terms called a hermitian matrix or is there any difference? Can we call the following matrix...

Why is it a (for example) 3x3 matrix of linear forms cannot necessarily be written as the sum of at most 3 rank one matrices of linear forms but the statement is true if "linear forms" is replaced with scalars? Does it have something to do with the 2x2 minors being calculated differently when...

Homework Statement Hi all. I am doing this work and can't seem to find any information on this in any of my notes or textbooks. The question is, "Evaluate (if possible) AB, BA, CD and DC", this is what i need some help with. I also have further on the question, "Evaluate | u |, | v |, u . v...

Homework Statement If A = \[ \left( \begin{array}{ccc} a & b \\ c & d \end{array} \right)\][\tex] and B=\[ \left( \begin{array}{ccc} \alpha & \beta \\ \gamma & \delta \end{array} \right)\] [\tex] in the basis |e1>,|e2>, find AxB (where "x" is the tensorproduct) in the basis...

The Pauli Spin matrices: \sigma_1=\left[ \begin{array}{ c c } 0 & 1 \\ 1 & 0 \end{array} \right],\sigma_2=\left[ \begin{array}{ c c } 0 & -i \\ i & 0 \end{array} \right],\sigma_3=\left[ \begin{array}{ c c } 1 & 0 \\ 0 & -1 \end{array} \right] are used...

Hello Just took on a new hobby..being game design know a little c# and decided to try my hand at xna...the problem is I am also trying my hand at vectors and matrices also, as I know this with gaming goes hand in hand. Kinda picking up the basics to vectors but with matrices not got a great...

why is S^n/S^m homotopic to S^n-m-1. the book just made this remark how do you see this geometrically. how do you compute fundamental groups of matrices like O(3) and SO(3) or SL(2) and whatnot.

Homework Statement exp^\prime(0)B=B for all n by n matrices B. Homework Equations exp(A)= \sum_{k=0}^\infty A^k/k! The Attempt at a Solution Obviously I want to calculate the limit of some series, but I don't know what series to calculate. I wanted to try \lim_{h \to...

Does anybody know a good thread, homepage or book that takes up different interpretations of Pauli and Dirac matrices with the connection to for example quaternions or bivectors? Maybe someone could comment on this?

I need help about similarity transformation in matrices. Is there anyone who knows how can I decide whether "the two matrices having the same eigenvalues" are similar or not without using eigenvectors? For example, following two matrices have the same characteristic polynomial. But they are...

Homework Statement [PLAIN]http://img530.imageshack.us/img530/6672/linn.jpg The Attempt at a Solution For parts (a) and (b) I've found the eigenvalues to be -\frac{1}{3} and -1 with corresponding eigenvectors \begin{bmatrix} -1 \\ 3 \end{bmatrix} and \begin{bmatrix} -1 \\ 1...

Hello. I am writing an encryption algorithm for a program and have decided to use a hill cipher. My problem is that for the hill cipher, I have to have matrices with very specific properties. How might I go about finding 3 matrices of size 4x4 such that all of the matrices are integer-only and...

[input_pattern, input_class]=loading(file_name,no_sample,no_vector); input_pattern and input_class are 60 X 60 matrices, how can I display these matrices ? Thank you.

Hello, I attached a copy of the problem and my attempted solution. The three Pauli spin matrices are given above the problem. I'm having trouble getting the right side to equal the left side, so I'm assuming I'm doing something wrong. When I got towards the end it just wasn't looking right...

So since I learning QFT a while ago, I've always struggled to understand fermions. I can do computations, but I feel at some level, something fundamental is missing in my understanding. The spinors encountered in QFT develop a lot from "objects that transform under the fundamental representation...

hi how to calculate the traces of product of Dirac matrices in QED. i want caculate crossection of process scattering in QED. a program to calculate it

I was surprised that I have never had to do this in so long and forgot the basic way to factor out a scalar multiple when a matrix is raised to a certain power (for example -1 for inverse matrices). Basically, I just want some confirmation: (λT)^n= λ^n (T^n ) ∶ for λ ϵ F and Tϵ L(V)...

Homework Statement I'm supposed to write a proof for the fact that det(A)=det(B) if A and B are similar matrices. Homework Equations Similar matrices have an invertible matrix P which satisfies the following formula: A=PBP^{-1} det(AB) = det(A)det(B) The Attempt at a Solution...

I try to understand how to calculate derivatives of functions, which contain matrices. For a start I am looking at derivatives by a single variable. I have x=f(t) and I want to calculate \frac{dx}{dt}. The caveat is that f contains matrices, that depend on t. Can I use the ordinary chain rule...

If A,B, and C are each nxn invertible matrices, will the product ABC be invertible?

Homework Statement Show that A = 3 4 3 -1 0 -1 1 2 3 is not diagonalizable but is triangulable and carry out triangulation (A has rational entries) I found that the only eigenvalue is 2, and that the characteristic equation is (x-2)3, but I'm...

Homework Statement Show that if A is nilpotent then det(I-A)=det(I+A)=1. Homework Equations I know that det(A)=0 if A is nilpotent and det(I)=1, so this seems like it follows logically. I also know that the tr(A)=0 and that tr(I-A)=tr(I+A)=n, and that the characteristic polynomial of...

Hi, I have a question about linear transformation. So given a matrix A in the basis u (denoted as A_u). Now in another basis that I don't know, A_u becomes A_v. How can I find v? (I know u, A_u and A_v). Thank you very much,

Homework Statement A biconconvex (n_l=1.5) lens have radii worth 20 and 10 cm and an axial width of 5 cm. Describe the image of an object whose height is 2.5 cm and situated at 8 cm from the first vertex.Homework Equations Transfer matrices. The Attempt at a Solution I used the ray transfer...

I have stumbled upon a problem which I have so far been unable to solve. I we consider a general set of linear equations: Ax=b, I know the the system is inconsistent which makes least square method the logical choice. So the mission is to minimize ||Ax-b|| And the usual way I do...

Homework Statement Let V=Rⁿ and let AεMnxn(R) Prove that <x,Ay> = <A^T x, y> for all x,yεV Homework Equations The Attempt at a Solution Can someone tell me if I'm on the right track? <x,Ay> = x^T Ay = (x^TA)y = <A^T x,Y>

If we consider the spin-1/2 pauli matrices it makes sense that [S_x,S^2] = [S_y,S^2] = [S_z,S^2] = 0 since S^2 = I... and this is supposed to be true in general, right? Well, if I attempt to commute the spin-1 pauli matrices given on http://en.wikipedia.org/wiki/Pauli_matrices, with...