Eigenvalues Definition and 820 Threads

1). suppose that y1, y2, y3 are the eigenvalues of a 3 by 3 matrix A, and suppose that u1, u2,u3 are corresponding eigenvectors. Prove that if { u1, u2, u3 } is a linearly independent set and if p(t) is the characteristic polynomial for A, then p(A) is the zero matrix. I thought...

i'm reading and doing some work in introduction to linear algebra fifth edition, and i came across some problems that i had no clue. 1. An (n x n) matrix A is a skew symmetric (A(transposed) = -A). Argue that an (n x n) skew-symmetrix matrix is singular when n is an odd integer. 2. Prove...

This thread, https://www.physicsforums.com/showthread.php?t=74810, was orignally posted here in the QM forum, but it was moved to the homework section, which is reasonable. But nobody there knows quantum mechanics. I guess the OP gave up on it, but I'm curious how to do the problem now. So if...

Hi. I have this problem which i am stuck at: Consider a one-dimensional Hamilton operator of the form H = \frac{P^2}{2M} - |v\rangle V \langle v| where the potential strength V is a postive constant and |v \rangle\langle v| is a normalised projector, \langle v|v \rangle = 1 ...

If L^2 |f> = k^2 |f>, where L is a linear operator, |f> is a function, and k is a scalar, does that mean that L|f> = +/- k |f>? How would you prove this?

In a recent thread https://www.physicsforums.com/showthread.php?t=67366 matt and cronxeh seemed to imply that we should all know that the product of the eigenvalues of a matrix equals its determinant. I don't remember hearing that very useful fact when I took linear algebra (except in the...

Hi, I need help on these questions for an assignment. I've been working on them for a couple of days and not getting anywhere. Any help would be appreciated... 1) A certain 4X4 real matrix is known to have these properties: 1. Two fo the eigenvalues of A are 3 and 2 2. the number 3 is an...

I having trouble finding the eigenvalues and eigenfunctions for the operator \hat{Q} = \frac{d^2}{d\phi^2}, where \phi is the azimuthal angle. The eigenfunctions are periodical, f(\phi) = f(\phi + 2\pi), which I think should put some restrictions on the eigenvalues. I think...

How can i find the eigen value(s) of A - (alpha)I where A is an arbitrary matrix ?

Hello: -was solving for the eigenvalues of a matrix. Obtained: \lambda = 1 \pm 2i -substituted back into matrix to try and solve for the eigenvectors: \left(\begin{array}{cc}2-2i & -2\\4 & -2-2i\end{array}\right) \left(\begin{array}{cc}x_1 \\ x_2 \end{array}\right) = \mathbf{0}...

let A be a diagonalizable matrix with eignvalues = x1, x2, ..., xn the characteristic polynomial of A is p (x) = a1 (x)^n + a2 (x)^n-1 + ...+an+1 show that inverse A = q (A) for some polynomial q of degree less than n

I'm having trouble getting started on this problem... I just really don't understand what to do. Solve X'+2X'+(\lambda-\alpha)X=0, 0<x<1 X(0)=0 X'(1)=0 a. Is \lambda=1+\alpha an eigenvalue? What is the corresponding eigenfunction? b. Find the equation that the other eigenvalues...

I'm trying to find the basis for a particular matrix and I get a 3 eigenvalues with two of them being identical to each other. What do I do to find the basis for the repeated eigenvalue? Will it have the same basis as the original number? Thanks!

I'm reading an introductionary text on quantum physics and am stumbling a bit with the terms used. The text discusses a finite potential box (one dimension, time independent). It calculates the conditions for the solutions of the wave functions, which I can follow perfectly. At that point...

let's say.. there is a particle, with mass m, in a 2-dimensions x-y plane. in a region 0 < x < 3L ; 0 < y < 2L how to calculate the energy eigenvalues and eigenfunctions of the particle? thx :smile: and.. 2nd question.. there is a particle of kinetic energy E is incident from...

Who knows the formula to calculate the eigenvalues of total angular momentum between two different states? In particular, what is the matrix element of <S, L, J, M_J | J^2 | S', L', J', M'_J> ? Thank's...

spanning sets, eigenvalues, eigenvectors etc... can anyone please explain to me what a spanning set is? I've been having some difficulty with this for a long time and my final exam is almost here. also, what are eigenvalues and eigenvectors? i know how to calculate them but i don't understand...

Find the eigenvalues and eigenvectors of the general real symmetric 2 x 2 matrix A= a b b c The two eigenvalues that I got are a-b and c-b. I got these values from this: (a-eigenvalue)(c-eigenvalue)-b^2=0 (a-eigenvalue)(c-eigenvalue) = b^2 (a-eigenvalue)= b = a-b...

I'm currently taking linear algebra and it has to be the worst math class EVER. It is extremely easy, but I find the lack of application discouraging. I really want to understand how the concepts arose and not simple memorize an algorithm to solve mindless operations, which are tedious. My...

the signature of a metric is often defined to be the number of positive eigenvalues minus negative eigenvalues of the metric. this definition has always seemed a little suspicious to me. eigenvalues are defined for endomorphisms of a linear space, whereas the metric is a bilinear functional...