Matrices Definition and 1000 Threads

Homework Statement Suppose a 2x2 matrix X (not necessarily hermitian, nor unitary) is written as X = a0 + sigma . a (the sigma . a is a dot product between sigma and a) where a0 and a1, a2 and a3 are numbers. How on Earth does X represent a matrix? it's a number added to another number...

Hey PF, I'm having trouble seeing the bigger picture here. Take matrix A and matrix B. If B can be obtained from A by elementary row operations then the two matrices are row equivalent. The only explanation my book gives is that since B was obtained by elementary row operations, (scalar...

Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that C'|v>=(A'+B')|v>=A'|v>+B'|v>. Now we can represent A',B',C' by matrices A,B,C respectively. I have a question about proving that if C'=A'+B', C=A+B holds. The proof is Using the above with Einstein summation...

Hi, can you guys recommend a book for an undergrad that uses matrices properties to explain statistics? Thx!

Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...

Homework Statement [/B] Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal. Show that the product <x, y> A = x · Ay is an inner product for R3. Homework Equations None The Attempt at a Solution I've shown that x · Dy is...

Homework Statement Suppose that ##s \to A(s) \subset \mathbb{M}_{33}(\mathbb{R})## is smooth and that ##A(s)## is antisymmetric for all ##s##. If ##Q_0 \in SO(3)##, show that the unique solution (which you may assume exists) to $$\dot{Q}(s) = A(s)Q(s), \quad Q(0) = Q_0$$ satisfies ##Q(s) \in...

Homework Statement Hello! Please, take a look at the problem described in the attached file. The question is: Explain why the transition matrix does what we want it to do.Homework Equations The Attempt at a Solution (sorry, I don't know yet how to type formulas) I don't quite understand this...

Homework Statement Hello! Here is the problem from Stitz-Zeager Pre-calculus book: At 9 PM, the temperature was 60F; at midnight, the temperature was 50F; and at 6 AM, the temperature was 70F . Use the technique in Example 8.2.3 to t a quadratic function to these data with the temperature...

Homework Statement I need to show that any normal matrix can be expressed as the sum of two commuting self adjoint matrices Homework Equations Normal matrix A: [A,A^\dagger]=0 Self Adjoint matrix: B=B^\dagger The Attempt at a Solution A is a normal matrix. I assume I can write...

This is a pretty basic question, but I haven't seen it dealt with in the texts that I have used. In the proof where it is shown that the product of a spinor and its Dirac conjugate is Lorentz invariant, it is assumed that the gamma matrix \gamma^0 is invariant under a Lorentz transformation. I...

I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence. On page 419 D&F define similar matrices as follows: They then state the following: BUT? ... how...

Hi All struggling with concepts involved here So I have $${P}_{2} = \left\{ a{t}^{2}+bt+c \mid a,b,c\epsilon R\right\}$$ is a real vector space with respect to the usual addition of polynomials and multiplication of a polynomial by a constant. I need to show that both...

Right, i don't believe this is a homework question. The only reason I am stating this is because PF are stringent with their rules. I'm quite confused and I'm not sure how to explicitly state my problem. The vertices of a triangle are (a,b) (c,d) and (e,f). This can be arranged into a...

Homework Statement Let R be the ring of all 2*2 matrices over Zp, p a prime,. Show that if det(a b c d) = ad - bc ≠ 0, then (a b c d) is invertible in R.Homework Equations The Attempt at a Solution I don't know how to start if Zp, with p a prime, is the clause. I know that since ad- bc ≠ 0, it...

Homework Statement Find the matrix representation of S_z in the S_x basis for spin 1/2. Homework Equations I have the Pauli matrices, and I also have the respective kets derived in each basis. There aren't really any relevant equations, other than the eigenvalue equations for the...

x=t^2-s^2, y=ts,u=x,v=-y a) compute derivative matrices \vec{D}f(x,y) = \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right] \vec{D}f(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] b) express (u,v) in terms of (t,s) f(u(x,y),v(x,y) = (t^2-s^2,-(ts)) c) Evaluate \vec{D}(u,v)...

Homework Statement Prove \exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\cosh \sqrt{\alpha^2+\beta^2}+\frac{\sinh \sqrt{\alpha^2+\beta^2}}{\sqrt{\alpha^2+\beta^2}}(\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x) Homework Equations e^{\hat{A}}=\hat{1}+\hat{A}+\frac{\hat{A}^2}{2!}+...

http://en.wikipedia.org/wiki/Gamma_matrices#Normalization See the image below. Which of us is right: me or Wikipedia?

The problem statement Let ##A ∈ K^{m×n}## and ##B ∈ K^{n×r}## Prove that min##\{rg(A),rg(B)\}≥rg(AB)≥rg(A)+rg(B)−n## My attempt at a solution (1) ##AB=(AB_1|...|AB_j|...|AB_r)## (##B_j## is the ##j-th## column of ##B##), I don't know if the following statement is correct: the columns of...

Homework Statement I have a function f:M_{n×n} \to M_{n×n} / f(X) = X^2. The questions Is valid the inverse function theorem for the identity matrix? It talks about the Jacobian at the identity, but I have no idea how get a Jacobian of that function. Can I see the matrices as vectors and...

Hi there! I'm back again with functions over matrices. I have a function f : M_{n\times n} \to M_{n\times n} / f(X) = X^2. Is valid the inverse function theorem for the Id matrix? It talks about the Jacobian at the Id, but I have no idea how get a Jacobian of that function. Can I see that...

1st: Not a specific problem, I just didn't know where else to put it. We just covered this today in class. Basically what we're doing is reducing higher level matrices to 2x2 matrices and using them to calculate the determinant. I asked my teacher where that came from, and he was really...

I am going through Mary Boas' "Mathematical Methods in the Physical Sciences 3rd Ed". I finished the chapter 3 section 12 problem set, but I do not understand how she gets eq. 12.39. These don't seem obviously equal to each other. Here is the equation: $$\lambda Tr=Vr$$ Where T is a matrix...

Is it possible to write series ##\ln x=\sum_na_nx^n##. I am asking you this because this is Taylor series around zero and ##\ln 0## is not defined. And if ##A## is matrix is it possible to write ##\ln A=\sum_na_nA^n##. Thanks for the answer!

Homework Statement Find two matrices E and F such that: EA= \begin{bmatrix} 2 & 1 & 2\\ 0 & 2 & 1\\ 0 & 3 & 0\\ \end{bmatrix} FA= \begin{bmatrix} 0 & 2 & 1\\ 0 & 3 & 0\\ 2 & 7 & 2\\ \end{bmatrix} Homework Equations The Attempt at a Solution So I know how to get...

Hi, i was wondering how the following expression can be decomposed: Let A=B°C, where B, C are rectangular random matrices and (°) denotes Hadamard product sign. Also, let (.) (.)H denote Hermitian transposition. Then, AH *A how can be decomposed in terms of B and C ?? For example, AH...

About the Newton's identities: I'm right if I state that ek = Ik, pk = tr(Ak) and hk = det(Ak) (being Ik the kth-invariant of the matrix A)?

Hi there! How can I prove that the space of matrices (2x2) nonzero determinant is dense in the space of matrices (2x2) ? I've already proved that it's an open set. Thanks. PD: Sorry about the mistake in the title.

Hi there! How can I prove that a function which takes an nxn matrix and returns that matrix cubed is a continuous function? Also, how can I analyze if the function is differenciable or not? About the continuity I took a generic matrix A and considered the matrix A + h, where h is a real...

Homework Statement For a system Ax= 0, suppose det (A)= .0001. Which of the following describes the solutions to system? There is exactly one solution, but the system is close to having infinity many. There is exactly one solution, but the system is close to having none. Homework Equations...

Homework Statement Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F_13 for i, j = 0,1,2, 3. Compute F(hat) and verify that F(hat)F = I Homework Equations The matrix F(hat) is called the inverse discrete Fourier transform of F. The Attempt at a Solution I found that e = 4...

Homework Statement (i) Verify that 5 is a primitive 4th root of unity in F13. (ii) Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F13 for i, j = 0,1,2, 3. Compute F(hat) and verify that F(hat)F= I. Homework Equations The matrix F(hat) is called the inverse discrete Fourier...

Homework Statement A=[a,1;0,a] B=[a,0;0,a] If I want to show if matrix A is NOT similar to matrix B. Is it enough to show that B=/=Inv(P)*A*P? Or would I need to show that they do not have both the same eigenvalues and corresponding eigenvectors?

Hello. I need some help with one question about relationship of two matrices. The task: Suppose that I is identity matrix, u - is vector, u' is transposed vector, α - real number. It can be prove that inverse matrix of I+α*u*u' has similar form I+x*u*u'. The task is to find x. I tried to...

Just curious, but the use of matrices isn't readily obvious to me, and I was wondering what a typical use would be?

Just trying to get my head around undergraduate quantum mechanics and they throw a lot of stuff at you, so some help is appreciated. I understand that the wave function is some abstract vector living in an infinite dimensional hilbert space, and that it's a function. But then the textbook I'm...

Is possible to rewrite the quadratic, cubic and quartic determinant in terms of matrices and matrix operations (trace and determinant)? https://en.wikipedia.org/wiki/Discriminant_of_a_polynomial#Formulas_for_low_degrees

Homework Statement The Attempt at a Solution I know what relations are individually but what do I do to represent the composition of both? Is it some matrix operation? Would I multiply them, but instead of adding I use the boolean sum?

Hi everybody.. How can i use fortran99 to find the eigenvalues & eigenvectors of sparse matrices? Thanx :)

Homework Statement If A and B are invertible matrices over an algebraically closed field k , show there exists \lambda \in k such that det(\lambda A + B) = 0 .The Attempt at a Solution Can anyone agree with the following short proof? I tried looking online for a confirmation, but I wasn't...

Homework Statement This makes intuitive sense to me, but I am getting stuck when trying to read the Dirac notation proof. Anyway, the author (Shankar) is just demonstrating that the product of two operators is equal to the product of the matrices representing the factors. Homework Equations...

1) Let A a square matrix, x a colum vector and b another colum vector. So, I want solve for x the following equation: Ax=b So: x=b÷A = b×A-1 And this is the answer! Or would be this the correct answer x = A-1×b ? 2) Is possible to solve the equation above for A ? How?

First off, I apologize if I'm in the wrong thread. I wasn't really sure where to put this. Alright, long winded question so stay with me (note: the actual question is at the end, so if you already know how to work it out, just skip ahead)! I was reading a math problem at the nsa.gov website (...

So let's say I have 2 matrices A and B. I need to solve 2 eqns involving specific elements of each matrix. e.g. A(1)+B(2)=4; A(1)-B(2)=2. Is there any way to do this? My efforts with Fsolve and solve have failed. Here's what I've done so far: function F=myfun(A,B)...

Homework Statement a) Find the inverse of the matrix: \begin{pmatrix}1 & 2 & 0\\ 2 & 0 & 1\\ 1 & 1 & 2\end{pmatrix} (sorry I don't know how to show a matrix more clearly on this) b) Write A and A-1 as a composition of matrices of the form Rij(k), Tij and D22(k) Homework...

Homework Statement I thought it would be better to attach it. Homework Equations The Attempt at a Solution So for the first part I've found that A^2=the Identity matrix, but from there I don't have much of an idea on how to substitute that into the equation for M and end up with...

Hello, The product of a 2x5 matrix P and a 5x3 matrix B shall be a 2x3 zero matrix. P and B are all matrices of integers. P = [6 2 -5 -6 1;3 6 1 -6 -5] One possible B is [0 -4 0;3 0 0;-1 -1 3;2 -3 -2;1 1 3] This solution B has a property: det(PPt) = det(BtB) = 7778 The question is: What...

Homework Statement A polynomial of degree two or less can be written on the form p(x) = a0 + a1x + a2x2. In standard basis {1, x, x2} the coordinates becomes p(x) = a0 + a1x + a2x2 equivalent to ##[p(x)]_s=\begin{pmatrix}a0\\ a1\\ a2 \end{pmatrix}##. Part a) If we replace x with...

I am trying to find the state equations for a mass spring system. I found the transfer function to be \[ H(s) = \frac{X_1(s)}{F(s)} = \frac{m_2s^2 + b_2s + k} {s\big[m_1m_2s^3 + (m_2b_1 + m_1b_2)s^2 +...