Eigenvalues Definition and 853 Threads

Given a 4x4 non-Hermitian matrix, is there any method I can use to prove the eigenvalues are real, aside from actually computing them? I'm looking for something like the converse of the statement "M is Hermitian implies M has real eigenvalues". When can one say that the eigenvalues of a...

Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix M from a matrix of its eigenvectors as columns P and a diagonal matrix of the eigenvalues E through M=P E P^{\dagger}?

Eigenvalues & Eigenvectors !SOLVED! Homework Statement Find the eigenvalues and eigenvectors of matrix A = \left( \begin{array}{cc} 2 & 2 \\ 3 & 1 \end{array} \right) Homework Equations Ax = \lambda x The Attempt at a Solution Solving \left\vert \begin{array}{cc} 2 - \lambda &...

Homework Statement Let B be an invertible matrix a.) Verify that B cannot have zero as an eigenvalue. b.) Verify that if \lambda is an eigenvalue of B, then \lambda^{-1}^ is an eigenvalue of B^{-1}. Homework Equations Bv = \lambdav, where v\neq0The Attempt at a Solution a.) I'm pretty sure...

Hello I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0. I can not find it anywhere =/ I think it was a physics teacher who told us this a couple of years ago, can anyone enlighten me? cheers

Hello Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues. Can someone please help.. Thanks

Homework Statement Let A and B be similar matrices a)Prove that A and B have the same eigenvalues Homework Equations None The Attempt at a Solution Firstly, i don't see how this can even be possible unless the matrices are exactly the same :S

Homework Statement Let A be an nxn matrix, and suppose A has n real eigenvalues lambda_1, ...lambda_n repeated according to multiplicities. Prove that det A = lambda_1...lambda_n Homework Equations None The Attempt at a Solution Could someone explain what is meant by 'repeated...

Homework Statement Let A be nxn matrix, suppose n has real eigenvalues,λ1,...,λn, repeated according to multipilicities. Prove that detA = λ1...λn. Homework Equations The Attempt at a Solution I started by applying the definition, Av = λv, where v is an eigenvector. then I just dun...

The real matrix A= \begin{pmatrix}\alpha & \beta \\ 1 & 0 \end{pmatrix} has distinct eigenvalues \lambda1 and \lambda2. If P= \begin{pmatrix}\lambda1 & \lambda2 \\ 1 & 0 \end{pmatrix}...

I have the Hamiltonian for an S=5/2 particle given by: H= a.Sz + b.Sz^2 +c.Sx where Sz and Sx are the spins in z and x directions respectively. The resulting matrix is tridiagonal symmetric but I can't find the eigenvalues..Any idea how to diagonalise it. N.B: a is a variable and must be...

Homework Statement Given X''(x) + lambda*X(x) = 0 X(0) = X'(0), X(pi) = X'(pi) Find all eigenvalues and eigenfunctions. Homework Equations Case lambda = 0 Case lambda > 0 Case lambda < 0 The Attempt at a Solution First case, X(x) = Ax + B but the function doesn't satisfy...

1. Find the General Solution of the given system [ -1 -1 2 ] X = X' [ -1 1 0 ] [ -1 0 1 ] det(A-lambda*Identity matrix) = 0, solve for eigenvalues/values of lambda (A-lambda*Identity matrix|0) The eigenvalues we got are 1 and 1 +/- i. The matrix generated for...

Homework Statement A is an invertible matrix, x is an eigenvector for A with an eiganvalue \lambda \neq0 Show that x is an eigenvector for A^-1 with eigenvalue \lambda^-1 Homework Equations Ax=\lambdax (A - I)x The Attempt at a Solution I know that I need to find x and then apply...

I am trying to get some intuition for Eigenvalues/Eigenvectors. One real-life application appears to be a representation of resonance. What are some practical uses for Eigenvalues? What other things may Eigenvalues represent?

Homework Statement Is there a basis of R4 consisting of eigenvectors for A matrix? If so, is the matrix A diagonalisable? Diagonalise A, if this is possible. If A is not diagonalisable because some eigenvalues are complex, then find a 'block' diagonalisation of A, involving a 2 × 2 block...

Homework Statement Let {u1, u2,...,un} be an orthonormal basis for Rn and let A be a linear combination of the rank 1 matrices u1u1T, u2u2T,...,ununT. If A = c1u1u1T + c2u2u2T + ... + cnununT show that A is a symmetric matrix with eigenvalues c1, c2,..., cn and that ui is an eigenvector...

Let V be the vector space of all real integrable functions on [0,1] with inner product <f,g>=\int_0^1 f(t)g(t)dt Three linear operators defined on this space are A=d/dt and B=t and C=1 so that Af=df/dt and Bf=tf and Cf=f I need to find the eigenvalues of these operators: For A...

I find it rather tedious to calculate the eigenvalues of a 3x3 matrix. For example The \emph{characteristic polynomial} $\chi(\lambda)$ of the 3$3 \times 3$~matrix \[ \left( \begin{array}{ccc} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array} \right)\] is given by the formula \[...

can somebody explain eigenvalues inside hilbert spaces??

I have 2 questions that need to be solve: 01. Find upper and lower bound for the k-th eigenvalue \lambda_{k} of the problem ((1+x^2)u')'-xu+\lambda(1+x^2)u for 0< x< 1 with boundary conditions u(0)=0 and u(1)=0 02. Find a lower bound for the lowest eigenvalue of the problem...

2. Construct a 3x3 matrix A that has eigenvalues 1, 2, and 4 with the associated eigenvectors [1 1 2]T, [2 1 -2]T and [2 2 1]T, respectively. [Hint: use P-1AP = K, where K is the diagonal matrix] hlp me... pls guild me to the step reli no idea how to do it

In a system of equations with several eigenvalues, what does it mean (signify) when one is strong (high in value) and the others are weak (low in value)? Can a general statement be made without referencing an application? If so, is there a math book that explains the idea?

hi I have the following eigenvalue problem -(x2y')'=λy for 1<x<2 y(1)=y(2)=0 I tried plugging an equation y=xa and you get the equation a2+a+λ=0 so for this I get that λ<1/4 to hava a solution. So does this mean, every λ smaller than 1/4 is an eigenvalue? do you know what else I...

Does anybody know a web page or a book, or the general method to find the eigenvalues and the eigenfunctions of laplacian u =lambda u inside the circle u=0 in the boundary thanks

Homework Statement Prove that a square matrix is invertible if and only if no eigenvalue is zero. Homework Equations The Attempt at a Solution If a matrix has an inverse then its determinant is not equal to 0. Eigenvalues form pivots in the matrix. If any of the pivots are...

Homework Statement find the eigenvalues of 3x3 matrix: I have to learn how to find eigenvalues of 3x3 matrix and this is the link, am I not supposed to do lamda-1 instead of 1-lamda like here? http://en.wikipedia.org/wiki/Eigenvalue_algorithm (the chapter name is "Eigenvalues of 3×3...

Hi there this is my first post here, I am having some trouble with a homework question in quantum. I need to normalize both ψ1 and ψ2 and then find the energy eigenvalues of each. The given Hamiltonian is H0 = (1 2 ) (2 -1) And ψ1 = (...

Let's say I have the equation p(t)f''(t)=Kf(t) with p(t) a known periodical function, K an unknown constant and f(t) the unknown function. This is an eigenvalues problem that once solved gives a set of K={k1, k2,...} eigenvalues. I get these eigenvalues and they coincide with the ones...

This question is about the use of eigenvalues in a specific application. The subject is Computer Vision and the topic is the Harris Corner detection method. The attached file is PDF document of slides that show the math in a bit more detail. In the slides, a corner is located by looking at...

Homework Statement Find the eigenvalues of B = [5 2 0 2], [3 2 1 0], [3 1 -2 4], [2 4 -1 2]. Compute the sum and product of eigenvalues and compare it with the trace and determinant of the matrix. Homework Equations The Attempt at a Solution I get the characteristic polynomial...

Background: I'm having trouble using principal component analysis to try and align two data sets. I have two sets of 3D point data, and I can use PCA to get principal axes of the two sets of data. I do this by finding the eigenvectors of the covariance matrix for each set of data. This gives...

Just wondering is there a way to get the characteristic equation of an n by n matrix without going through tedious calculations of solving multiple determinants of matrices?

Homework Statement compute and plot the 10 lowest energy eigenvalues of a particleinan infinity deep spherically symmetric square well? Homework Equations The Attempt at a Solution

Hi How can i compute(or obtain from mathmatical tables) and plot a several energy eigenvalues of a particle in an infinity deep ,spherically and symmetric square well?

Homework Statement Hi all. I am given by following linear system: \begin{array}{l} \dot x = dx/dt = ax \\ \dot y = dy/dt = - y \\ \end{array} The eigenvalue of the matrix of this system determines the stability of the fixpoint (0,0): A=\left( {\begin{array}{*{20}c} a & 0 \\...

Homework Statement In the Leslie population model, suppose matrix A has a strictly dominant eigenvalue \lambda_1. Age class evolution is given by: x^{(k)} = Ax^{(k-1)}; initial population is x^{(0)}. (i) Initially, let x^{(0)} be the linear combination a_1x_1 + a_2x_2 + ... + a_nx_n, of A's...

Am I understanding this right? Let's say I have a 15x15 matrix called Z. Then the matrix of eigenvalues calculated from Z, called D, can have two forms - either diagonal or block diagonal. If the matrix D comes out with values only on the diagonal, then there are only real values. But...

Homework Statement I am studying about eigenvalues and norms. I was wondering whether the way I understand them is correct. Homework Equations The Attempt at a Solution The eigenvalue of a matrix those that satisfy Ax = \lambda x, where A is a matrix, x is an eigenvector, \lambda...

Homework Statement C is an operator that changes a function to its complex conjugate a) Determine whether C is hermitian or not b) Find the eigenvalues of C c) Determine if eigenfunctions form a complete set and have orthogonality. d) Why is the expected value of a squared hermitian...

can somebody help me with the solution of the following problems? Ques. Find the eigenfunctions and eigenvalues for the operators 1. sin d/d psi 2. cos(i d/d psi) 3. exp(i a d/d psi) 4. (d)square/d (x)square+z/x * d/dx

Homework Statement What are the eigenvalues of the set of operators (L1^2, L1z, L2^2, L2z) corresponding to the product eigenstate \left\langlem1 l1 | m2 l2 \right\rangle? PS: If you have Liboff's quantum book, this is problem #9.30. Homework Equations We've also been learning...

It seems to be true, that if some eigenvalue of a Hamilton's operator is an isolated eigenvalue (part of discrete spectrum, not of continuous spectrum), then the corresponding eigenstate is normalizable, and on the other hand, if some eigenvalue of a Hamilton's operator is not isolated, then the...

Hello, Is it sufficient to determine that a nXn matrix is not diagonalizable by showing that the number of its distinct eigenvalues is less than n? Thanks for your time.

Hello! First of all let me wish you a happy new year! This is not a homework problem, but rather a curiosity of mine. In Schwabl's Quantum Mechanics, one can find the proof of the fact that all eigenvalues of the angular momentum Lz are either integers or half-integers, raging from -l to l(l...

my question is take A= {(5,0,-1),(2,3,-1),(4,0,1)} find all eigenvalues and eigenvectors by using the characteristic equation i get -(lamda-3)3 however its the next bit i don't understand, in the answers (A-3I)(x,y,z)=(0,0,0) is used which I'm perfectly ok with and then (A-3I)2 is used and...

g(x,y) = x^3 - 3x^2 + 5xy -7y^2 Hessian Matrix = 6x-6******5 5********-7 Now I have to find the eigenvalues of this matrix, so I end up with the equation (where a = lambda) (6x - 6 - a)(-7 - a) - 25 = 0 Multiplying out I get: a^2 - 6xa + 13a - 42x + 17 = 0 How am I supposed to solve...

Matrix A= 1 2 0 2 1 0 2 -1 3 i got eigenvalues k=3 k=-1 what do i do after that to prove it is not able to be diagonalized

Homework Statement Hey guys, for my linear algebra class I need to find the signs of the eigenvalues (I just need to know how many are positive and how many are negative) of an nxn matrix with zeros everywhere except for the two diagonals directly above and directly below the main diagonal...

Hi, everyone! While I was studying for my midterm, I encountered this question. ------ Consider the hermitian operator H that has the property that H4 = 1 What are the eigenvalues of the operator H? What are the eigenvalues if H is not restricted to being Hermitian? ------ What I am...