Matrices Definition and 1000 Threads

Whenever a problem seems too easy, I assume I'm missing something :-) This is in a section on Legendre polynomials ... Given the series $ \alpha_0 + \alpha_2Cos^2\theta +\alpha_4C^4 +\alpha_6C^6 = a_0P_0 + a_2P_2 + a_4P_4 +a_6P_6 $ (abbreviating $Cos^n\theta$ to $C^n$) Express both...

I thought I had this clear, then I met operators and - at least to me - the new information overlapped with, and potentially changed, that understanding. Research on the web didn't help as there seem to be different uses & opinions ... So what I am trying to do is NOT make a summary of what...

Create a matrix with 4 rows and 8 columns with every element equal to 0. Create a second, smaller matrix with 2 rows and 4 columns where each row is [1 2 3 4]. Replace the 0s in the upper left-hand corner of the bigger matrix with the values from the smaller matrix. (If you do this correctly...

Is there any chart/graph/website online or in a ebook that has a clear concise list of special matrices used in physics? I'm just getting into an intro to quantum mechanics class and we are going over all types of matrices, Identity, hermitian, diagonal, transpose, unitary, and so on. I want...

Homework Statement This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of...

The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices. I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix. The answer given is here, relevant answer is (b): Imgur link: http://i.imgur.com/DKwt8cN.png...

I'm taking a Differential Equations class and we're dealing with matrices and determinants. I've dealt with them before but I was always annoyed by the fact that I don't know what the heck is going on. So I know that matrices are a way to organise linear equations and make transformations...

Homework Statement Find an example of two unitary matrices that when summed together are not unitary. Homework EquationsThe Attempt at a Solution A = \begin{pmatrix} 0 & -i\\ i & 0\\ \end{pmatrix} B = \begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix} A+B = A = \begin{pmatrix} 0 & 1-i\\ 1+i &...

Homework Statement Find all 2x2 Matrices which are both hermitian and unitary. Homework Equations Conditions for Matrix A: A=A^† A^†A=I I = the identity matrix † = hermitian conjugateThe Attempt at a Solution 1. We see by the conditions that A^† = A and by the second condition, we see that...

Homework Statement Let K and L be symmetric PSD matrices of size N*N, with all entries in [0,1]. Let i be any number in 1...N and K’, L’ be two new symmetric PSD matrices, each with only row i and column i different from K and L. I would like to obtain an upper bound of the equation below...

Homework Statement Show that the product of two nxn unitary matrices is unitary. Is the same true of the sum of two nxn unitary matrices? Homework Equations Unitary if A†A=I Where † = hermitian conjugate I = identity matrix. The Attempt at a Solution [/B] We have the condition: (AB)†(AB)=I I...

Homework Statement Show that the sum of two nxn Hermitian matrices is Hermitian.Homework Equations Hermitian conjugate means that you take the complex conjugate of the elements and transpose the matrix. I will denote it with a †. I will denote the complex conjugate with a *. The Attempt at a...

Homework Statement Find all diagonal unitary matrices. Homework Equations The Attempt at a Solution I think I am starting to get the hang of this type of material. I hope I am right in my thinking. So if we have a diagonal matrix, let's say a 2x2 for a simple example: \begin{pmatrix} a &...

Homework Statement Show that |A_ij| ≤ 1 for every entry A_ij of a Unitary Matrix A. Homework Equations A matrix is unitary when A^†*A=I Where † is the hermitian operator, meaning you Transpose and take the complex conjugate and I = the identity matrix The Attempt at a Solution I'm having a...

Homework Statement Show that (A+B)*=A*+B* Homework Equations I think I am missing a property to prove this. The Attempt at a Solution This should be easier then I am making it out to be. But I seem to be missing one key property to do this. A*+B* is just A(ij)*+B(ij)* = Right hand side...

Homework Statement Let ##G=GL_n(F)## for ##F## a field, and define an equivalance relation by ##A\sim B## iff ##A## and ##B## are conjugate, that is, iff ##A=PBP^{-1}## for some ##P\in GL_n(F)##. Does ##\sim## respect multiplication?Homework Equations The equivalency respects multiplication...

Good afternoon all, I'm taking a linear algebra course this semester, and upon entering the topic of 'Applications of Matrix Operations', my professor has given our class the opportunity to earn some extra credit points by writing a paragraph or two on the application of stochastic matrices in...

A very small country town has a population that can be grouped according to three categories: adults teenagers and children. Each year statistics show that: Children are born at the rate of 4% of the adult population 12% of children become teenagers 15% of teenagers become adults 0.5% of...

Homework Statement Assuming I understand the problem correctly, I need to define the set of all orthogonal matrices. Homework Equations The Attempt at a Solution Per the definition of orthogonal matrix: Matrix ##A\in Mat_n(\mathbb{R})## is orthogonal if ##A^tA = I## If ##O## is the set of all...

Homework Statement Find all 2 x 2 and 3 x 3 orthogonal matrices which are diagonal. Construct an example of a 3 x 3 orthogonal matrix which is not diagonal. Homework Equations Diagonal Matrix = All components are 0 except for the diagonal, for a 2x2 matrix, this would mean components a and d...

$$A$$ is a hermitian matrix with eigenvalues +1 and -1. Let $$\left|+\right>$$ and $$\left|-\right>$$ be the eigenvector of $$A$$ with respect to eigenvalue +1 and eigenvalue -1 respectively. Therefore, $$P_{+} = \left|+\right>\left<+\right|$$ is the projection matrix with respect to eigenvalue...

Homework Statement Let A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} Find all 2 x 2 matrices B such that AB = BA. Homework Equations http://euclid.colorado.edu/~roymd/m3130/Exam2sol.pdf The Attempt at a Solution I let B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and set AB=BA...

MIT OCW 18.06 using Intro to Linear Algebra by Strang So I was working through some stuff about Cyclic Matrices, and the text was talking about how the column vectors that make up this cyclic matrix, shown here, are coplanar, and that is the reason that Ax = b will have either infinite...

Homework Statement I'm a bit at a loss - I thought the last row with '1's would be useful, but it just gave me: (b2c - bc2) - (a2c - ac2) + (a2b - ab2) and bc(b - c) - ac(a - c) + ab(a - b) But then it is a dead end. I am probably doing something stupid again ... Any help appreciated.

Homework Statement Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)} a) Show that B1 is a basis for [R][/3] b) Find the coordinates of w=(2,3,1) relative to B1 c)Given that B2 is a basis for [R[/3], find...

Hey all. I am currently reading an article and there is a paragraph that I am having a hard time understand. This is what the paragraph says: "Since Ar = Arτ and Ai = -Aiτ, we know that only the lower triangular (including the diagonal) elements of Ar are independent and only the strictly lower...

I have just started to study quantum mechanics, so I have some doubts. 1) if I consider the base given by the eigenstates of s_z s_z | \pm >=\pm \frac{\hbar}{2} |\pm> the spin operators are represented by the matrices s_x= \frac {\hbar}{2} (|+><-|+|-><+|) s_y= i \frac...

1. Get A from its inverse3. I believe it has something to do with the theorem that states: E1E2E3...EkA=I

Hi all, Firstly, I am not sure whether this is the area of the forum to ask this. I have been learning and researching a completely different topic, and from this I have come across a completely new concept of the Kronecker function. I have done a google search on this to get the intro and...

I have two questions, but the second is only worth asking if the answer to the first is yes: Are the spin matrices for three particles, with the same spin, σ ⊗ I ⊗ I, I ⊗ σ ⊗ I and I ⊗ I ⊗ σ for particles 1, 2 and 3 respectively, where σ is the spin matrix for a single one of the particles? I...

Asked to determine the eigenvalues and eigenvectors common to both of these matrices of \Omega=\begin{bmatrix}1 &0 &1 \\ 0& 0 &0 \\ 1& 0 & 1\end{bmatrix} and \Lambda=\begin{bmatrix}2 &1 &1 \\ 1& 0 &-1 \\ 1& -1 & 2\end{bmatrix} and then to verify under a unitary transformation that both can...

Folks, What is the idea or physical significance of simultaneous diagonalisation? I cannot think of anything other than playing a role in efficient computation algorithms? Thanks

Homework Statement [/B] What are the ##n\times n## matrices over a field ##K## such that ##M^2 = 0 ## ? Homework Equations The Attempt at a Solution Please can you tell me if this is correct, it looks ok to me but I have some doubts. I have reused the ideas that I found in a proof about...

Homework Statement Show that ##n\times n ## complex matrices such that ##\forall 1\le i \le n,\quad \sum_{k\neq i} |a_{ik}| < |a_{ii}|##, are invertible Homework EquationsThe Attempt at a Solution If I show that the column vectors are linearly independent, then the matrix has rank ##n## and...

Hi all, This isn't actually part of my assigned homework, I was just trying it out as the topic confuses me. I think I might understand what's going on a little more if someone could walk me through this. Any advice on the intuition behind it would be great. Thanks so much. 1. Homework...

Consider the rotation group ##SO(3)##. I know that ##R_{x}(\phi) R_{z}(\theta) - R_{z}(\theta) R_{x} (\phi)## is a commutator? But can this be called a commutator ##R_{z}(\delta \theta) R_{x}(\delta \phi) R_{z}^{-1}(\delta \theta) R_{x}^{-1} (\delta \phi)##?

Homework Statement Are the following matrices hermitian, anti-hermitian or neither a) x^2 b) x p = x (hbar/i) (d/dx) Homework EquationsThe Attempt at a Solution For a) I assume it is hermitian because it is just x^2 and you can just move it to get from <f|x^2 g> to <f x^2|g> but I am not...

Homework Statement Look at the matrix: A = sin t sin p s_x + sin t sin p s_y +cos t s_z where s_i are the pauli matrices a) Find the eigenvalues and normalized eigenvectors (are they orthogonal)? b) Write the eigenvector of s_x with positive eigenvalue as a linear combination of the...

Hello, I have been studying matrices and determinants recently and do not understand why certain things are done the way they are. Like, why is matrix multiplication defined the way it is. I find that there are not enough proofs. Is there any book/article/video that any of you recommend to...

Hello, Here's a textbook question and my solution, please check if it is correct, I'm slightly doubtful about the second part. Consider Hermitian matrices M_1, M_2, M_3,\ and\ M_4 that obey: M_i M_j+M_j M_i = 2 \delta_{ij} I \hspace{10mm} for\ i,\ j\ = 1,\ ... ,4 (1) Show that the eigenvalues...

For some reason I just can't seem to wrap my head around the idea of reducing a Matrix to row echelon form. I'm familiar with the steps that the textbooks and tutorials use and how it's done but when I try practicing on my own I feel lost. e.g. all I end up with are just a bunch of random...

Homework Statement http://www.math.harvard.edu/archive/21b_spring_09/faq.html I'm having trouble understanding this explanation, particularly this part. "The Kyle numbers are 1, 2 because adding the first to 2 times the second column gives zero. " Sorry for this basic question but I was...

Homework Statement Basically, derive the formula ## m = \frac{ 25 cm}{f_e} \frac{L}{f_o} ## using ray matrices. This just has variable tube length and assumes eye to object distance is 25 cm. Homework Equations Ray matrices: ## \left[ \begin{array}{cc} 1 & d \\ 0 & 1 \end{array} \right] ##...

I'm trying to learn column space currently and I'm confused about the meaning of rows and columns. So I'm given this definition for column space: "The column space of matrix A is the set Col A of all linear combinations of the columns of A" Given the matrix A: [ 1 -3 -4 ] [ -4 6 -2 ] [ -3 7...

Homework Statement Consider the bases B = {b1,b2} and B' = {b'1,b'2} for R2, where b1=(1, -1), b2=(2,0), and b'1=(1,2), b'2=(1,-3) a. Find the transition matrix P from B to B' b. Compute the coordinate matrix [p]B, where p=(4,3); then use the transition matrix P to compute [p]B' Homework...

Homework Statement How many 3x2 matrices are there in RREF form? Homework EquationsThe Attempt at a Solution I counted 5, but the solutoin in my book says 4. 0 0 0 0 0 01 0 0 1 0 0 1 1 0 0 0 00 1 0 0 0 0 1 0 0 0 0 0

Consider the system $x'_1 = -5x_1 + 1x_2$ $x'_2 = 4x_1 - 2x_2$ If we write in matrix form as $X' = AX$ then a) X = b) X' = c) A = d) Find the eigenvalues of A. e) Find eigenvectors associated with each eigenvalue. Indicate which eigenvector goes with which eigenvalue. f) Write the...

Hi there! As you might have already guessed, I'm referring primarily to the 'geometrical' difference (is there such geometry in Hilbert space?) between ##n##-dimensional state vectors | \psi \rangle = \left( \begin{matrix} \psi_1 \\ \psi_2 \\ \vdots \\ \psi_n \end{matrix} \right) and their...

Homework Statement For which ##2x2## matrices ##A## does there exist an invertible matrix ##S## such that ##AS=SD##, where ##D= \begin{bmatrix} 2 & 0\\ 0 & 3 \end{bmatrix} ##? Give your answers in terms of the eigenvalues of ##A##. Homework Equations ##A\lambda=\lambda\vec{v}## The Attempt...

I have been reading through Mark Srednicki's QFT book because it seems to be well regarded here at Physics Forums. He discusses the Dirac Equation very early on, and then demonstrates that squaring the Hamiltonian will, in fact, return momentum eigenstates in the form of the momentum-energy...