Matrices Definition and 1000 Threads

One last question on these topics, I need to choose the WRONG statement, and they all seem correct to me... a) If A is a squared matrix for which \[A^{2}-A=0\] then A=0 or A=i b) If A and B are diagonal matrices, then Ab=BA c) A 4X4 matrix with eigenvalues 1,0,-1,2 is "diagonlizable" d) The...

Homework Statement Let A be the set of all 3x3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices B in A for which the system of linear equations B \left[ \begin{array}{c} x \\ y \\ z \end{array} \right] =...

Hey folks. I'm working on a project which seems to be encountering a problem. I took Linear Algebra a few years ago in college, and haven't really applied it very much so I'm at a bit of a loss here. I have a solution to a iterative nonlinear least squares problem: Trilateration with n...

I'm trying to find a general solution for the logistic ODE \frac{dU}{dx}=A(I-U)U, where A and U are square matrices and x is a scalar parameter. Inspired by the scalar equivalent I guessed that U=(I+e^{-Ax})^{-1} is a valid solution; however, U=(I+e^{-Ax+B})^{-1} is not when U and A don't...

Hi all, I have two matrices A=0 0 1 0 0 0 0 1 a b a b c d c d and B=0 0 0 0 0 0 0 0 0 0 a b 0 0 c d I need to prove that two eigenvalues of A and two eigenvalues of B are equal. I tried to take the determinant of A-λI...

Could anyone let me know of a good resource that I could use to learn matrix mathematics? I'm not sure if that is the proper term for that segment of mathematics, but hopefully you get the gist of it. It can be a book or a website, does not matter to me. Also, any suggestions as to what I...

I need to prove that a 3x3 matrix with all odd entries will have a determinant that is a multiple of 4. This is how I set it up: I let A = { {a, b, c}, {d, e, f}, {g, h, i} } with all odd entries then I define B = { {a, b, c}, {d + na, e + nb, f + nc}, {g + ma, h + bm, i + cm} } where I add...

Homework Statement 2. The attempt at a solution So part a. makes sense to me, it basically comes down to A1 = 1 -1 -1 -1 1 1 -1 1 1 A2 = 1 -1 -1 -1 2 -2 -1 -2 11 I'm not sure how to approach part b. because the question doesn't make much sense to me...

Matrix proof :( 1. Let A and B be two similar matrices. characteristic in the space λ is an eigen value, show that : sized V_λ^A = sized V_λ^B 2. Let A invertible matrix. A ∈ ℝ nxn and invertible matrix ⇔ 0, A is not an eigen value. 3. Let A and B be two similar matrices...

##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R}) is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.

Hello I have been trying to solve a couple of true / false questions, and I am not sure my answers are correct, I would appreciate it if you could verify it. The first question is: A and B are matrices such that it is possible to calculate: \[C=AB+B^{t}A^{t}\] a. A and B are of the same...

An operator A defined by a matrix can be written as something like: A = Ʃi,jlei><ejl <eilAlej> How does this representation translate to a continuous basis, e.g. position basis, where operators are not matrices but rather differential operators etc. Can we still write for e.g. the kinetic...

Lets suppose a 4×4 matrix A has two identical rows with some other 4×4 matrix B. Does that imply there determinant is equal? Or does it really say nothing about how the determinants of the two matrices are related.

Hi, In QFT we define the projection operators: \begin{equation} P_{\pm} = \frac{1}{2} ( 1 \pm \gamma^5) \end{equation} and define the left- and right-handed parts of the Dirac spinor as: \begin{align} \psi_R & = P_+ \psi \\ \psi_L & = P_- \psi \end{align} I was wondering if the left- and...

As far as I know there is no explicit formulas but is this true? I've tested it in Matlab with random matrices and It seems true! cond(A+B) =< cond(A) + cond(B) Where can I find a proof for this hypothesis?

Dear PhysicsForum, We have just treated the Dirac equation and its lagrangian during our QFT course, but we have only gone in depth in 3+1 dimensions. My question is about what happens to spin in 2+1 dimensions. In 3+1 dimensions we have to use 4 by 4 gamma matrices, but in 2+1 dimensions we...

aklsdk

Homework Statement Whats up guys! I've got this question typed up in Word cos I reckon its faster: http://imageshack.com/a/img5/2286/br30.jpg Homework Equations I don't know of any The Attempt at a Solution I don't know where to start! can u guys help me out please? Thanks!

for fixed ## m \geq 2 ## let ## \epsilon (i,j) ## denote the mxm matrix ## \epsilon (i,j)_{rs} = \delta _{ir} \delta _{js} ## when m = 2000 show by formal calculations that i) ## \epsilon (500,199) \epsilon (1999,10) = \epsilon (500,10) ## ii) ## \epsilon (1999,10) \epsilon (500,1999) = 0##...

Hi everyone, :) I have very limited knowledge on linear algebra and things like Jordan Normal form of matrices. However I am currently doing an Advanced Linear Algebra course which is compulsory and I am trying hard to understand the content which is quite difficult for me. One of the things...

Please do not be offended by my literary style. I find thinking about mathematical problems in such a way helps me learn better. A is a 2x2 matrix of complex numbers, call this "apple" B is a 2x2 matrix of complex numbers, call this "banana" Let a "Fruit Salad" be defined as follows: S...

I need to determine the matrix that represents the following rotation of $R^3$. (a) angle $\theta$, the axis $e_2$ (b) angle $2\pi/3$, axis contains the vector $(1,1,1)^t$ (c) angle $\pi/2$, axis contains the vector $(1,1,0)^t$ Now, I would like to check if I got the right answers because...

1. Consider the 2x2 matrix \sigma^{\mu}=(1,\sigma_{i}) where \sigma^{\mu}=(1,\sigma) where 1 is the identity matrix and \sigma_{i} the pauli matrices. Show with a direct calcuation that detX=x^{\mu}x_{\mu} 3. I'm not sure how to attempt this at all...

I posted this question over at the QM page, https://www.physicsforums.com/showthread.php?t=714076 but I realized I am really looking for a hard Mathematical proof ... A description of a numerical way of proving this would also be very helpful for me. or a reference covering the...

Just going over my linear algebra notes and I've forgotten the formal definition of ## \epsilon(i,j)_{rs} ## I have written down ## \epsilon (i,j)_{rs} = \delta_{ir}\delta_{js} ## but I can't seem to remember what r and s represent. Also, I don't quite understand why it equals ##...

Hi, Wasn't sure if I should post this to Linear Algebra or here. My question is really simple: Can a 2N by 2N random, and Hermitian Matrix ( Hamiltonian ) be always written as: H = A \otimes I_{2\times 2} + B \otimes \sigma_x + C \otimes \sigma_y + D \otimes \sigma_z where A,B,C,D are all...

True or False. If true explain or prove answer, and if false give an example to show the statement is not always true. 1. If A is a 4x4 matrix and a1+a2=a3+2a4, then A must be singular. 2. If A is row equivalent to both B and C, then A is row equivalent to B+C. My Work: 1. I say it's...

Homework Statement Find all the values of h and k such that the system: hx + 6y = 2 x + (h+1)y = k has: (a) No solutions (b) A unique solution (c) Infinitely many solutions Homework Equations - The Attempt at a Solution I've tried putting the system into echelon form and got...

Homework Statement Here is the problem: http://img801.imageshack.us/img801/6770/oaza.png Homework Equations None really, just gauss jordon elimination I assume unless I am missing out on something The Attempt at a Solution First I multiplied the first row by -5 then added...

Hi everyone, I have two matrices A and B, A=[0 0 1 0; 0 0 0 1; a b a b; c d c d] and B=[0 0 0 0; 0 0 0 0; 0 0 a b; 0 0 c d]. I have to proves theoretically that two of the eigenvalues of A and B are equal and remaining two eigenvalues of A are 1,1. I tried it by calculating the...

I read that scalar matrices are the center of the ring of matrices. How would I prove this? Tips are appreciated. It is already obvious that scalar matrices commute with all matrices, but the converse seems tricky. BiP

Homework Statement Find h so that: -8x + -7y = 7 16x + hy = 14 has infinitely many solutions (solve this exercise with matrices). Homework Equations - The Attempt at a Solution I converted the system to matrix form, but when I try to convert it to echelon form, I get the...

Hi, The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get. I have, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix. I understand the equation...

Trying to make sense of the following relation: \sum log d_{j} = tr log(D) with D being a diagonalized matrix. Seems to imply the log of a diagonal matrix is the log of each element along the diagonal. Having a hard time convincing myself that is true, though

The commutator of two operators A and B, which measures the degree of incompatibility between A and B, is AB - BA (at least according to one textbook I have). But multiplying/substracting matrices just yields matrices! (http://en.wikipedia.org/wiki/Matrix_multiplication). So firstly, how can a...

I am trying to prove that for any two vectors x,y in ##ℂ^{n}## the product ## \langle x,y \rangle = xAy^{*} ## is an inner product where ##A## is an ##n \times n## Hermitian matrix. This is actually a generalized problem I created out of a simpler textbook problem so I am not even sure if this...

Suppose that ##A## is a diagonalizable ## n \times n ## matrix. Then it is similar to a diagonal matrix ##B##. My question is, is ##B## the only diagonal matrix to which ##A## is similar? I have thought about this, but am unsure if my answer is correct. My claim is that ##B## is the only...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Homework Statement If A is positive definite, show that ## A = C C^T ## where ## C ## has orthogonal columns. The Attempt at a Solution So, I've got the first part figured out. Because ## A ## is symmetric, an orthogonal matrix ## P ## exists such that ## P^TA P = D =...

Homework Statement Express the product where σy and σz are the other two Pauli matrices defined above. Homework Equations The Attempt at a Solution I'm not sure if this is a trick question, because right away both exponentials combine to give 1, where the result is...

I was wondering if anybody can see where I have gone wrong here? I was given the rule, E2(E1A1)=(E2E1)A1, I can't seem to find my mistake.

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Hi, I think I need a sanity check, because I've been working on this for a while and I can't see what I'm doing wrong! According to several authors, including Sakurai (Modern QM eq 3.3.21), a general way to write an operator from SU(2) is...

Hello, I have the following matrix of matrices \mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array} where each element is a square matrix, A is a diagonal matrix of real numbers, whereas B is not (necessarily), and the superscript H means conjugate transpose...

Suppose ##A## is a ## n \times n## matrix. Define the set ## V = \{ B | AB = BA, B \in M_{n \times n}( \mathbb{F}) \} ## I know that ##V## is a subspace of ##M_{n \times n}( \mathbb{F}) ## but how might I go about finding the dimension of ##V##? Is this even possible? It seems like an...

I have two real symmetric matrices A and B with the following additional properties. I would like to know how the eigenvalues of the product AB, is related to those of A and B? In particular what is \mathrm{trace}(AB)? A contains only 0s on its diagonal. Off diagonal terms are either 0 or...

If two matrices are similar, it can be proved that their determinants are equal. What about the converse? I don't think it is true, but could someone help me cook up a counterexample? How does one prove that two matrices are not similar? Thanks! BiP

If I have two square nonnegative primitive matrices where the Perron-Frobenius Theorem applies how would I calculate lim (A^k)(B^k) as k approaches infinity.

I am having trouble getting the kraus matrices(E_k)) from a unitary matrix. This task is trivial if one uses dirac notation. But supposing I was coding, I can't put in bras and kets in my code so I need a systematic way of getting kraus matrices from a unitary matrix(merely using matrices). So...

I'm confused about encoding matrices. In my textbook, the encoding matrix H was given as the matrix such that if U is a codeword then HUT is the zero vector. In this case, the number of columns in H would be the length of the codeword. But in another explanation, I read that an encoding matrix...