Matrices Definition and 1000 Threads

A,B are nxn. If AB is invertible. Show that A and B are invertible. I know how to prove it by determinant, using linear transformations and contradictions. I am looking for a direct way using a proof by matrices. Can anyone think of one? Thank you.

Show the following: If A and B are n x n matrices such that A - B is singular then A2 - AB + BA - B2 is also singular. I really have no clue how to solve this, but I am guessing that AB does not equal BA, I don't know how that can help or be relevant but just in case Thanks alot...

"Pauli matrices with two spacetime indices" Hi all. This is my first post so forgive me if my latex doesn't show up correctly. I am familiar with defining a zeroth Pauli matrix as the 2x2 identity matrix to construct a four-vector of 2x2 matrices, $\sigma^\mu$. I'm trying to read a paper...

Homework Statement By taking the trace of both sides prove that there are no finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p,x] = -ih/2pi Why does this argument fail if the matrices are infinite dimensional? Homework...

Homework Statement Ok so I have to construct a real symmetric matrix R whose eigenvalues are 2,1,-2 and who corresponding normalized eigenvectors are bla bla bla.. So let the matrix of eigenvalues down diagonal be E and matrix of eigen vectors be V Is R = VEV^T or R = V^TEV?? How...

If I have a matrix for which all eigenvalues are zero, what can be said about its properties? If I multiply two such matrices, will the product also have all zero eigenvalues? Thanks

Hi, I have a bunch of images saved as .fig and I need to obtain the data in these files. I tried something as simple as image1 = imread('figure1.fig'); but it does not recognize the format. If anyone could help me out that would be greatly appreciated, thanks!

in this system of matrices, ignoring the N M matrix, is matrix [A B B D] equal to [C] given that the matrices represent a system of linear equations in the form Ax = b? I'm just wanting to know whether it's like algebra, where you can divide both sides by the same thing and keep equality...

Homework Statement How do we prove that commuting matrices have common eigenvalues? Homework Equations The Attempt at a Solution

I need help proving the identity \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}=\gamma^{\mu}g^{\nu\rho}+\gamma^{\rho}g^{\mu\nu}-\gamma^{\nu}g^{\mu\rho}+i\epsilon^{\sigma\mu\nu\rho}\gamma_{\sigma}\gamma^{5}

Homework Statement Find the energy eigenvalues and eigenfunctions for the one-dimensional infinite square well. Calculate the matrices for the position and momentum operators, Q and P, using these eigenfunctions as a basis.Homework Equations The energy eigenvalues are E_n = \frac{\pi^2...

Hi, I'm trying to write an SOR program in Matlab and have everything done, except I cannot figure out how to create a matrix of the following form without manually typing everything in: x=(1,0,...,0,1)T for various sized matrices. Thank you for your help.

Homework Statement (C-CB)^{-1}=B^{-1}E Solve the system for B, with the assumption that C,B, and (C-CB) are invertible. Homework Equations The rules for matrix invertibility (but I've already come to the conclusion that all matrices in this equation are invertible. The Attempt at a...

What is the purpose of diagonalisation of matrices? Why do teach this stuff? Is there any serious tangible application of diagonalisation? Do engineers or physics need this process?

Homework Statement The Attempt at a Solution I think I first need to find T(e2)=? and T(e2)=? and then combine those into a matrix. I am having trouble starting to solve for T(e1) and T(e2) so far I have [1] = alpha [1] + beta [3] [0] [2]...

Homework Statement Let A be a square matrix with right inverse B. To prove A has a left inverse C and that B = C. Homework Equations Matrix multiplication is asociative (AB)C=A(BC). A has a right inverse B such that AB = I The Attempt at a Solution I don't really know where to...

I try to understand if I am calculating the derivatives correctly or if I do something wrong. Here is an example: f(t)=xT*eAt*B*x t is a scalar, x is a vector, A,B are square matrices. df/dt=xT*A*eAt*B*x Is this correct?

Homework Statement Given: x (dot) y = x^T * y (where x,y are vectors; dot is dot product; and x^T is x transpose) and R is an orthogonal nxn matrix, and x,y are elements of R^n Show ||Rx|| = ||x|| The Attempt at a Solution I'm not sure what information I am suppose to use to solve...

Hi, I get a lot of questions about calculating M^k, where M is a square matrix! They say you can use an equation like M^k=PD(P^-1) where D is a diagonal matrix. I don't know how to calculate this! Any help will be appreciated! P.S. Sorry if this is in the wrong section!

I'll start off with my question: Why do we use Gaussian Elimination when inverting a matrix? (this is only one of the methods...which is the only one that doesn't make sense to me). I know how to do it, but I'm not sure why it works. When solving a system of linear equations, I understand...

Here is my problem. Any ideas are appreciated. Let P be a projection matrix (symmetric, idempotent, positive semidefinite with 0 or 1 eigenvalues). For example, P = X*inv(X'*X)*X' where X is a regressor matrix in a least square problem. Let A be a symmetric real matrix with only integer...

Figured it out.

Homework Statement Given the set of 3x3 matrices of the form: [1, a, b; 0, 1, c; 0, 0, 1], where a, b, and c are any real numbers show that the inverses of these matrices are of the same given form. Homework Equations Using elementary row operations, transform [A:I] into [I:A-1]. Inverse of...

I have created a program in MatLab doing LU factorization and need to implement a routine, so that MatLab automatically runs the program on all of my matrices. I have 6 matrices A1, A2, A3, A4, A5 and A6. But for the time being I can only run the program for one matrix, write the result down...

Homework Statement Most invertible matrices can be written as a product A=LU of a lower triangular matrix L and an upper triangular matrix U, where in addition all diagonal entries of U are 1. a. Prove uniqueness, that is, prove that there is at most one way to write A as a product. b...

Homework Statement By using the general density matrix rho find the average of the three Pauli matrices. You can then tell how many independent experiments you must make in order to determine rho. Homework Equations The Attempt at a Solution I know the Pauli matrices and their...

Homework Statement Prove that for any m x s matrix A and any s x n matrix B it holds that: rank(A) + rank(B) - s is less or equal to: rank(AB) The Attempt at a Solution Obviously, the following are true: - rank(A) is less or equal to s, - rank(B) is less or equal to s, -...

Homework Statement Find the image location of point (5,2) after reflection in the x-axis followed by rotation through 180 degrees about the origin. Homework Equations Matrix Transformation The Attempt at a Solution None, need help!

Inverse of a sum of matrices [solved] The problem is relatively simple. Given the equation: (I+2A)^{-1}= \begin{bmatrix} -1 & 2 \\ 4 & 5 \end{bmatrix} Find A. My problem seems to be that I'm distributing the inverse on the LHS incorrectly. My real question then is, is the...

Homework Statement Find the components of A after a rotation of -45 degrees about X3. A=(1,1,2) Homework Equations \lambda= (cos\theta 0 -sin\theta) ( 0 1...

Good day to all, I am stuck with this. I am trying to construct a matrices with this properties... if n = 8, suppose the matrix with size 1 by 8 become [16 -16 16 -16 16 -16 16 -16] if n the matrix become [2n -2n 2n -2n 2n -2n 2n -2n ] with size 1 by n I do appreciate if someone...

Hi, thank you for viewing this thread. My question is as follow: Suppose A is a n x m matrix and B is a m x n matrix, and we also know that the matrix B has infinite solutions, then what will the solution/s of the matix product AB be? I am thinking that it might be a matrix of infinite...

In Zee's quantum theory text, introducing the Dirac equation, he states the gamma matrices as direct products of Pauli matrices. The statements involve the identity matrix, sigma matrices, and tau matrices. It took me a bit to realize that the latter were identical. I hadn't seen the tau...

(a)Determine the row rank of the matrix, 1 1 1 1 1 1 2 5 2 2 0 -6 (b) What is the column rank of this matrix? (c) What is the dimension of the solution space Mx=0 So this is my answer: I have reduced my matrix into echelon form and i...

Homework Statement Show that (I-A)^{-1} = I + A + A^2 + A^3 if A^4=0 The Attempt at a Solution I found at Google Books some kind of formula for it...

Dear guys, I know that gamma matrices have some relations, like \gamma^0{\gamma^\mu}^\dagger\gamma^0 = \gamma^\mu \quad---(*) And I am wondering if this is representation independent? Consider, S\gamma^0S^{-1}S{\gamma^\mu}^\dagger S^{-1}S\gamma^0 S^{-1} = S\gamma^\mu S^{-1}...

Homework Statement "How many 5x5 permutation matrices are there? Are they linearly independent? Do they span the space of all 5x5 matrices?"Homework EquationsThe Attempt at a Solution The first two questions are fairly easy. 5! = 120 P matrices. Since dim(space of all 5x5 matrices) = 25...

Why do we need to diagonalise a matrix? What purpose does it serve apart from finding the powers of a matrix? Is there any tangible application of this?

I have to teach myself pre-calculus and basic calculus over the summer, and whilst covering matrices the chapter on solving simultaneous systems of equations using matrices puts forth several methods, one of which being the method of Gaussian elimination with augmented matrices. I understand why...

At present I introduce matrices as an array of numbers and then carry out various matrix operations. Is there a more tangible way of introducing this topic? I have thought of transformations but my experience with students has been that they get lost in the transformations and so give up on...

Homework Statement Given two 4x4 Matrices A = [0 -1 1 1, -1 1 0 0, 0 0 -1 1, 0 0 0 0] B = [-0.5 -0.5 -0.5 -1.5, -0.5 1.5 0.5 -0.5, 0 0 -1 1, 0 0 0 0] I need to show that these two matrices are similar. Homework Equations A = SBS^-1 which simplifies to AS = SB The Attempt...

Homework Statement Let A \in M_{n x n }(F) and let \gamma be an ordered basis for F^{n} . Then [L_{A}]_{\gamma} = Q^{-1} A Q , where Q is the n x n matrix whose j-th column is the j-th vector of gamma. The Attempt at a Solution I think I'm confused about some of the technical...

I googled for a proof,but didn't find one. Could anyone give me a link to a proof?

Homework Statement find all matrices x that satisfy the given matrix equation [ 1 2 3 4 5 6] * X = I_2 I_2 is the identity matrix 2x2 Homework Equations The Attempt at a Solution I just inverted the square matrix [ 1 2 4 5] so it becomes [5 -2 -4 1 ] so X should be [ 5 -2 -4 1 0...

Hi all, I'm interested in proving/demonstrating/understanding why the Dirac gamma matrices, plus the associated tensor and identity, form a complete basis for 4\times4 matrices. In my basic QFT course, the Dirac matrices were introduced via the Dirac equation, and we proved various...

I'm wondering if anybody could suggest some techniques that might be brought to bear on the following problem: Suppose a finite sequence M_1,M_2,\dots,M_k of 4\times 4 orthogonal reflection matrices is given. I'm interested in determining conditions on these matrices that will guarantee that...

This is the question: What must fulfill a matrix to be invertible in module Zn? Demonstrate. Z refers to integers. I really appreciate that someone could help me with this because i couldn't find strong information about it. I think that considering A as a matrix... the det(A) must be coprime...

Homework Statement Is the set of invertible 3x3 matrices a subspace of 3x3 matrices? Homework Equations The Attempt at a Solution I think no - the 'neutral 0 element' is not in the subset since the 3x3 0 matrix is not in the subset. Am I right? The book says it's not a subspace...

Doing some quantum mechanics, I just ran into the notion of a matrix whose elements have matrix values for the first time. Specifically, a 2x2 matrix whose elements are 4x4 matrices. This got me wondering how I can extend the question into the absurd. I can't think of any good reason that...

For example, a esemble of 50% spin up and 50% spin down electrons, the other of 25% spin up,25% spin down, 25% x direction spin up and 25% x direction spin down. The density matrix is identity matrix for both(correct me if I'm wrong), is there any way to distinguish the two experimentally?