Matrices Definition and 1000 Threads

I am a bit confused about the page that I attached... I don't understand part (ii)... How can you compute X[T]X? So why is T(v_1)=v_2 and T(v_2)=av_1 + bv_2? Thanks in advance...

Hi. Define a linear mapping F: M2-->M2 by F(X)=AX-XA for a matrix A, and find a basis for the nullspace and the vectorspace(not sure if this is the term in english). Then I want to show that dim N(F)=dim V(F)=2 for all A, A≠λI, for some real λ. F(A)=F(E)=0, so A and E belongs to the nullspace...

(I am not sure if this is the right section for this). This question probably is extremely trivial and silly, but I haven't been able to find the answer to it despite going through quite a bit of The Internet. So, it appears that each Quantum Logic Gate corresponds to a matrix. Ones that...

Let T : R2 -> R2 and S : R2 -> R2 be linear transformations defined by: T(x; y) = (5x + y ; 2x + 2y) and S(x; y) = (3x + 2y ; x): (i). Find the image of the line 2x + 3y = 5 under T. (ii). Find the natural matrices of the linear transformations T o S and T^-1 Sorry, I haven't done...

Homework Statement Show that if A, B and A + B are all invertible and the same size then A(A-1 + B-1)B(A + B)-1 = I And what does the result say about A-1 + B-1 The Attempt at a Solution I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by...

If A and B are similar matrices, show that A has an inverse IFF B is invertible. A=P^-1 * B * P Where P is an invertible matrix. (A)^-1 = (P^-1 * B * P)^-1 A^-1 = P^-1 * B^-1 * P Does this show what the question wanted?

Homework Statement If A≥0 and Ak>0 for some k≥1, show that A has a positive eigenvector. Homework Equations The Attempt at a Solution A is nxn Well from a previous problem we know that the spectral radius ρ(A)>0 We also know that if A≥0, then ρ(A) is an eigenvalue of A and...

I have one more question, I have two matrices A and B, both squared and with the same order. And I have a scalar, a not equal to zero. why are these statements not correct ? 1. If A and B are invertible, then a\cdot (B^{-1}A^{-1}B)^{^{t}} is not necessarily invertible 2. If A and B are...

I now know that inverses are only defined for square matrices. My question is: is this because inverses for non-square matrices do not exist, i.e. there is no (m by n) matrix A for which there exists an (n by m) matrix B such that both AB = I and BA = I is true? Or is it just done for...

Homework Statement Write down a basis for the space of nxn symmetric matrices. The Attempt at a Solution I just need to know what the notation for this sort of thing is. I understand what the basis looks like, and I was even able to calculate that it would have dimension...

Homework Statement Find the rank off matrices? i)A=[2 0 9 2; 1 4 6 0; 3 5 7 1 ] 3X4 ii)A=[3 1 4; 0 5 8; -3 4 4; 1 2 4;] 4X3 Find Eigen Vectors and Values of A; A = [3 2 4; 2 0 2; 4 2 3 ] Homework Equations -when det(A) is not equal to zero it will the rank of matrices...

Homework Statement I have read the following text in a textbook(look the attaxhement) ,and i have a simple question .WHY every 2x2 hermitian matrix would have to satisfy this Equation.It is not obvious to me why.Does anyone know the answer? The textbook stops there without giving any...

Normal Matrices Examples (URGENT) I need to produce 2 x 2 normal matrices A and B such that A + B is not normal. I have proven that AB is normal if AB = BA using the Householder matrix form. But I can't find a form for A + B failing to be normal. A matrix A with entries a, b, c, d is going...

Homework Statement

Why all operators in QM have a Hermitian Matrices ?

Have data that is coming from Matlab and want to read it into Fortran. Simple example of what I have: PROGRAM file_read IMPLICIT NONE REAL, ALLOCATABLE :: read_matrix(:,:) INTEGER :: i,j OPEN(1,FILE = 'data_wanted.prn', ACTION = 'READ', STATUS = 'OLD') do i = 1,50 do j = 1,25 READ(1,*)...

How do you find matrices a,b,c satisfying a=b*c*b^-1 b=c*a*c^-1 c=a*b*a^-1 ?

If I have (for simplicity) a vector ( A, B) where A and B are matrices how does the transpose of this look, is it ( AT, BT) or (AT BT)

Homework Statement Find (X * Y-1)T - (Y * X-1)T When X = [3 5] .....[1 2] and Y = [3 4] ...[2 3] Homework Equations Inverse= 1/ad-bc [d -b] ......[-c a] The Attempt at a Solution I got: [9 -6 ] [14 -9] But the answer is: [-3 -2] [6 3]I did the problem twice and got the same answer so I...

Ignore post, I found a counterexample to (2). I'm studying for an upcoming exam, and I'm a bit confused about how to go about proving or disproving the statement (2). 1.) Products of diagonalizable matrices are never diagonalizable. I figured false and my counterexample is really just the...

Homework Statement Let ej denote the jth unit column that contains a 1 in the jth position and zeros everywhere else. For a general matrix An×n, describe the following products. (a) Aej (c) eTiAej? Homework Equations Rows and Columns of a Product Suppose that A = [aij] is m × p and B =...

I am curious about under what conditions the powers of a square matrix can equal the identity matrix. Suppose that A is a square matrix so that A^{2} = I At first I conjectured that A is also an identity matrix, but I found a counterexample to this. I noticed that the counterexample...

Hello, Would it be correct to say that if for every two different vectors x and y, A*x ≠ A*y (where A is a symmetrical matrix), then A is NOT necessarily invertible? In other words, albeit for any two different vectors x and y symmetrical matrix A times one of the vectors is not equal to A...

How can I turn a 5x5 matrix into a 4x4? I really cannot remember and I need to do it in a coursework I am doing :/ I have a handout on how to do 4x4 into 3x3 but the handout is very confusing.

Hi, I have a bunch of closed differential equations that I want to solve. The variables of the DEs are 2x2 matrices. So, I want to enter some 2x2 matrices of variables and then use NDSolve to get the solution. How should I define a 2x2 matrix with four variables inside it? I tried...

Homework Statement Let A and B be two square matrices of order n such that AB = A and BA = B. Then, what is the value of [(A + B)t]2? Homework Equations The Attempt at a Solution [(A + B)t]2 = AtAt + AtBt + BtAt + BtBt. I tried to use the fact that AB = A and BA = B to keep...

As part of a larger problem involving classifying intertwining operators of two group representations, I came across the following question: If X is an n \times n diagonal matrix with n distinct non-zero eigenvalues, then exactly which n \times n matrices A satisfy the following equality...

I am trying to investigate the statistical variance of the eigenvalues of sample covariance matrices using Matlab. To clarify, each sample covariance matrix, \hat{\mathbb{R}}_{nn}, is constructed from a finite number, N, of vector snapshots, each sized (L_{vec} \times 1) (afflicted with random...

Homework Statement Consider the network of streets with intersections a,b,c,d and e below. The arrows indicate the direction of traffic flow along the oneway streets, and the numbers refer to the exact number of cars observed to enter or leave a,b,c,d and e during one minute. Each xi denotes...

Homework Statement I have a 4x4 matrix V composed of complex numbers. I also have a 1x4 matrix S. The question asks to solve for c in S = Vc Homework Equations I learned that c = <V, S>/||V||^2, or c= (1/a)*<V, S> where "a" is the value of the entries on the main diagonal of the...

Hello, I am new to this: Taking the first Pauli Matrix: σ1=[0 1 1 0] Doing the transpose it becomes: [0 1 1 0] So is it a unitary matrix? Similarly σ2= [0 -i i 0] Doing a transpose =[0 i [-i 0] Does it mean the complex conjugates are...

I have a nxn tridiagonal matrix (let's name it A) and i want to find a way to solve Ap, p=1,2,3,...inf, most efficient* (using the structure of my matrix) my first problem is how many calculations do i need for A2, and then how many calculations for the hole Ap ? any help please!*by most...

What is the exact definition for equivalent matrices? Is it necessary that it should be A = B if A,B are two matrices? Thanks a lot.

Hi I have a question regarding notation of matrices. I am trying to conserve space in my report, so instead of writing my matrix fully like this \left( {\begin{array}{*{20}c} 1 & 2 \\ 0 & { - 5} \\ \end{array} } \right) my plan is to write it as (1,0 ; 2 -5)^T. Is this notation...

This thread is posted to examine the proposition that all matrices define linear transformations. But what of the matrix equation? \left[ {\begin{array}{*{20}{c}} 0 & 1 & 0 \\ \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {blue} \\ {red} \\ {green} \\...

|1 | |4| |0 | |5| |1| |6 | A|-1 |= |5| ' A |-1|= |3| and A |1|= |8 | |0 | |0| | 1| |5| |1| |11| The first question is, determine the dimensions of A. So I can tell it is a 3x3 Then I'm asked to determine the columns of A, I'm not sure about...

Generally, Gamma matrices with one lower and one upper indices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally construct gamma matrices with two lower indices. There...

My book introduces the concept of adjacency and incidence matrices but I don't understand its use. Normally we shift from mathematical symbols and representation to graphical interpretation like in Cartesian graphs - to visualize functions better we draw them on a graph. But here we are doing...

Hi, I was wondering if someone could check my work for this linear algebra problem. I have attached the problem statement in the file "problem" and my work in the file "work." I would type out my work on here, but I couldn't figure out how to put matrices in a post so I just took a pic of my...

I'm looking for a determinant formula for a 10 by 10 matrices in variable format.

Homework Statement Just a matter of convention (question) Homework Equations \gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix} The Attempt at a Solution If then, \gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0...

Homework Statement Solve the equation. What is it's trace?Homework Equations k γμ γ5 o γ\nu γ5 The Attempt at a Solution I don't think this is reduced enough. γμkμγ5γ\nuo\nuγ\nuγ5 trace: just got rid of gamma5 with anticommutation. -Tr[γμkμγ\nuo\nuγ\nu]

\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0 How do I expand i\hbar \gamma^0 the matrix in this term, I am a bit lost. All the help would be appreciated!

Let A and B be n × n matrices. a. Show that if A is invertible and AB = 0, then B = 0. If A is invertible, it can be reduced to the I matrix. Thus IB=0 (this is the part where I'm hesitant, can I say that IB=0?) Thus B=0 since I≠0

Homework Statement I need help to expand some matrices Homework Equations \pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0 The Attempt at a Solution How do I expand i\hbar \gamma^0 the matrix in this term, I am a bit lost. All the help would be...

Say I have a mxn matrix A and a nxk matrix B. How do you prove that the matrix C = AB is full-rank, as well?

I have figured out the answer to the question, but I have no idea why and how it works. I have attached a copy of the question. I do apologize I am still having trouble putting into latex, I can install some but not all, so bare with me. So if I multiple out the matrices I get \chi2 +...

Can anyone help me in proving the following identity: (\gamma ^{\mu} )^T = \gamma ^0 \gamma ^{\mu} \gamma ^0 I understand that one can proceed by proving it say in standard representation and then proving that it's invariant under unitary transformations. this last thing is the one...

Hi I'm trying to figure out the proof of why the gamma matrices are traceless. I found a proof at wikipedia under 'trace identities' here http://en.wikipedia.org/wiki/Gamma_matrices (it's the 0'th identity) and from the clifford algebra relation \{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu \nu}...

Hi! I'm reading David Tong's notes on QFT and I'm now reading on the chapter on the dirac equation http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf and I stumbled across a statement where he claims that (\gamma^0)^2 = 1 \ \ \Rightarrow \text{real eigenvalues} while (\gamma^i)^2 = -1 \...