Linear Algebra and Group Theory

Hello, I have a linear algebra problem that I need help with. Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large matrices (6^n x 6^n, where n>= 3, to be specific). Currently, we are just using MATLAB's eig() function to get them. I am trying to find optimizations for the simulations to cut down on computing time. There are three matrices that we use. H_constant - generated before the loop. Real and symmetric about the diagonal. Does not change after initial calculation. H_location - generated during each iteration. Diagonal. H_final - H_constant + H_location. Therefore, it is also real and s…

I am trying to work through multiple Schaums outline books on my own including Schaum's Outline of Differential Geometry. I cannot seem to get how to change basis in 3D vectors. From chapter 1, problem 1.47, Given vectors u1, u2, and u3 form a basis in E3 and v1 = -u1 + u2 -u3, v2 = u1 +2u2 - u3, v3 = 2u1 + u3, show that v1, v2, v3 are linearly independent and find the components of a = 2u1 - u3 in terms of v1, v2, and v3. END OF PROBLEM. Now that v1, v2, v3 are linearly independent is obvious since there is a zero coefficient for u2 in the equation for v3, and I can show that the determinate of the matrix formed by the row vectors (-1, 1, -1), (1,…

Hi, This is the question that needs a proofs, as follows: Show that the smallest element of a nonempty subset of [math]\mathbf{W}[/math] is unique. My attempt at the proof, as follows: Let [math]\mathbf{U} \subseteq \mathbf{W}[/math], by the well ordering principle (WOP) we have that [math]a \in \mathbf{U}[/math] such that [math]a \leq x [/math] [math]\forall x \in \mathbf{U}[/math]. Now suppose [math]b \in \mathbf{U}[/math] such that [math]b \leq x [/math] [math]\forall x \in \mathbf{U}[/math]. Since [math]0 \leq x - a[/math] and [math]0 \leq x - b[/math] by definition. Now, [math]0 \leq x + x - (a + b)[/math] [math]a+ b \leq x + x = 1…

Hey, I just have a small question regarding the conjugation of permutation groups. Two permutations are conjugates iff they have the same cycle structure. However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate permutations by s My question is, how can you find out how many conjugation permutations (s) are within a group which also conjugate a and b. So for example (1 4 2)(3 5) conjugates to (1 2 4)(3 5) under s = (2 4), how could you find the number of alternate s's in the group of permutations with 5 objects? Would it be like (1 4 2) (3 5) is the same as (2 1 4) (35) which …

Here the question I need to prove followed by my attempt at the solution; [math] \bigcup_{\beta \in \mathcal{B}} A_{\beta} \subseteq \bigcup_{\alpha \in \mathcal{A}} A_{\alpha} [/math] and suppose [math] \mathcal{B} \subseteq \mathcal{A} [/math] My attempt at the solution, as follows: Let [math] x \in \bigcup_{\beta \in \mathcal{B}} A_{\beta} [/math] such that for some [math] (\beta \in \mathcal{B}) [/math] we have [math] x \in A_{\beta} [/math]. Now, Pick [math] \beta \in \mathcal{B} [/math] , since [math] \mathcal{B} \subseteq \mathcal{A} [/math] we have [math] \beta \in \mathcal{A} [/math]. Hence we have [math] x \in A_{\alpha} [/math] for some …

Dato numero, invenire quot modis multangulus esse possit, or Given a number, to find in how many ways it can be polygonal. I am looking for historical solutions to this problem of Diophantus, whether old or recent. I have found that Fermat implied he had a solution, but he gave no actual solution. (as if. lol) Any help is appreciated. Thanks.

Hi, i should be this proof Let [latex]R[/latex] be an ordering of [latex]A[/latex]. Prove that [latex]R^{-1}[/latex] is also an ordering of [latex]A[/latex], and for [latex]B\subset{A}[/latex], (A) [latex]a[/latex] is the least element of [latex]B[/latex] in [latex]R^{-1}[/latex] if and only if [latex]a[/latex] is the greatest element of [latex]B[/latex] in [latex]R[/latex]. (B) Similarly for (minimal and maximal) and (supremum and infimum. Good look!

I hope someone could explain how to calculate the following derivatives, Assume A, B are kxk full rank matrices I is identity matrix of the same order and τ is a nonnegative scalar variable and λ real number then define the following two functions φ(τ) = ln(det(τΑ+λΙ)) ψ(τ) = tr((τΑ+λΙ)-1)B which are the first and second derivatives of φ(.) and ψ(.) w.r.t. the argument τ? A proof and/or references should be usefull. Thank you

please tell me the no. of generators of the groups of order 60.

x = x can be termed as one of the most basic and fundamental law ofarithematical mathematics. This law can easily be proved as. Let x be not equal to x. then let x=1 and x=2 but 1 is not equal to 2 (logics) hence, x=x. Don't you feel the power of this equation? Even of it's simplitiy, if thisproved wrong, will thorougly devastate the present treatment of mathematic,making each and every constant power to attain a variablic value. x=x and x+ function=y : x = y or x is not equal to y made the whole arithematic mathematics. We build on these two basic axiomsand made what see as a mathematics. Logical Deduction. Logical Deduction were building blocks of …

Hello all, I'm seeking help in logical equivalence for the following: {[(p^q)V(p^r)]V(q^r)}^~[(p^q)^r] It's asking to have an equivalence of that equation using only negations of conditionals(biconditionals too I assume because of the v) First I take (p∧q), (p∧r), and (q∧r) and (p∧q) which are all conjunctions and change them into the negation of a biconditional based on ^ being the same as <-> which is a conditional so that they are equivalent. ~(p->~q), ~(p->~r), ~(q->~r), ~(p->~q) which gives me the new equation of {[~(p->~q)v~(p->~r)v~(q->~r)]}∧~[~(p->~q)∧r] I then take the remaining ∧ which are {[~(p->~q)v…

find the linear transformation which maps (1,0,1) to (1,1) (1,1,1) to (0,1) & (1,0,0) to (1,0)? please explain.

I typically run a simulation that draws a line between two random point in 3d space. then I throw two random numbers that represent two lines ( see red lines in attachment) starting from the two previuos points and sit on top of the original line. if the lengths of those two lines are such that they cross I update a register. I repeat that J times. I want to generalize this to higher than 3d. Is that possible and is it unique. Please provide any links to such information. Please check this thread, the last post, it says the concept of line (or I guess he means line intersection ) does not work in higer than 3d. I wonder why. My link Thank you for …

hi all i need some help with somthing for my robotics hobby. heres the problem: i have a robot. it has a scanning distance sensor.(sharp ir gp2 20-150 cm to those who are interested.) now what i want to do is i need an expression to tell me if a gap is large enough to fit through. the example demonstrates this. i have the sharp ir mounted on a servo (motor) the sharp ir turns. you have acess to the following pieces of info: variables: distance1 (you can take this at anytime the sharp is rotating. i can code it to take it at the edge of the wall.) distance2: same as distance 1 angle 1: taken at the same time as distance 1. it is the angel of the servo at…

hi so, i need to understand linear vector space and how it is useful in quantum mechanics...i dont know where to post it...basically can u suggest a good buk for me..which explains the concepts clearly

Let a matrix of a system of linear equations be W and a matrix that it is multiplied by as P. The order of coefficients corresponding to the variables for matrix W would be X, Y, and Z distributed across the row as X for the first column, Y for the second column, and Z for the third; however, this is ordered differently for the matrix of P, where the value of X, Y, and Z are distributed across in rows; where X is for the first row, Y is for the second, and Z is for the third. Is there a reason for the difference in order, or that there is really no reason to it and it is done for the sake of order? In short, why are multiplications of matrices ordered so that the dime…

So my textbook says multiply one of the linear equation until it can cancel out with another linear equation to get the solution for a variable, when the two equations are added. My question is, what is the proof or justification for adding equations that arrives to a solution? In other words, is this a mere coincidence that by adding the two equations, we can get the answer for a variable?

Does the matrix[latex]\left[ \begin{array}{cc} x&y \end{array} \right][/latex][latex]\left[ \begin{array}{ccc} a&b&c\\d&e&f\\k&m&n \end{array} \right][/latex][latex]\left[ \begin{array}{c} x\\y\\1 \end{array} \right][/latex] has a name? I believe it opens to become [latex]{a{x^2}+e{y^2}+(b+d){xy}+(k+c){x}+(f+m){y}+n}[/latex]

Hi everyone, I have tried to figure this out myself but I couldn't find any useful references. Does anyone have a clue? Does any Lie group G based on the simple Lie algebras An, Bn, Cn, Dn, E6, E7, E8, F4, G2 on the complex numbers have the following properties: i) it's simply connected, or it coincides with its universal covering group; ii) there exists an irrep r with dimension < dim(G) (i.e. number of generators) such that r (and r*) are contained in the composition r x r* Ideally r should be the fundamental, hence the *, but if this holds with a real irrep smaller than the adjoint it would also do. For SU(n) this does not work, the first irr…

This is a rather unusual "theoretical" question. I was thinking of basic arithmetic..of a certain kind and was wondering if there is (probably) an area of theo maths or physics that covers it. The way we view the world as far as counting is concerned is that there are 2 ways to go, up or down. For example from number 3 i get to 4 by adding 1 or i get to 2 by subtracting 1. This is the definition of addition in the simplest form. (subtraction can be argued is addition of negative numbers) Is there a 'space' where there are 3 ways to go, so that from element 'a' we can go to 'b', 'c' or 'd'? Adding 1 wouldn't really make sense then, or it would have to be defin…

I am fixing to start college and I need the best allround scientific calculator. I will probably be doing a little Algebra Geometry and Trig. What would ya suggest?

Hi, I would like some help with this, Assume A and B nxn matrices 1. If A,B are positive definite matrices what about the sign of tr(AB)? 2. If A is positive definite and B is positive semi definite what about the sign of tr(AB)? Thanks!

hey frnds...plzz help me in finding applications of group theory used in real life.....plzz help me....

The baby-step-giant-step method to compute the number m of rational points over an elliptic curve defined over the finite field \mathbb{F}_p Uploaded with ImageShack.us In the second part R=(p+1)P, but for every point on the curve (p+1)P is the identity element of the group: P_{\infty}. So R+iQ is always iQ, isn't it?

Dear all, I came across the following equation involving an upper triangular matrx: MR + R'M' = 2I, where M is a given pxp real-valued matrix, R is an unknown pxp real-valued upper triangular matrix with strictly positive entries on the diagonal and I is a pxp identity matrix. The prime (') denotes matrix transpose. I verified directly for the cases that p = 2 and p = 3 that the solution does not exist always and that the closed-form component-wise solutions are rather cumbersome. I should add that the entries of the matrix M are limited by |Mij| <= 1. Does anyone know the conditions under which the above equation admits a solution for R and of any clos…