Eigenvalues Definition and 820 Threads

Homework Statement Let A = \left[ \begin{array}{cc} -6 & 0.25 \\ 7 & -3 \end{array} \right] Find an invertible S and a diagonal D such that S^{-1}AS=D Homework Equations I basically have the question answered, just ONE problem.The Attempt at a Solution My answer is...

Homework Statement Prove a symmetric (2x2) matrix always has real eigenvalues. The problem shows the matrix as {(a,b),(b,d)}. Homework Equations The problem says to use the quadratic formula. The Attempt at a Solution From the determinant I get (a-l)(d-l) - b^2 = 0 which...

Homework Statement How would I go about proving that if a linear operator T\colon V\to V has all eigenvalues equal to 0, then T must be nilpotent? The Attempt at a Solution I know that this follows trivially from the Cayley-Hamilton theorem (the characteristic polynomial is x^n and hence...

Homework Statement Let B = (1 1 / -1 1) That is a 2x2 matrix with (1 1) on the first row and (-1 1) on the second. Homework Equations The Attempt at a Solution A) (1 1 / -1 1)(x / y) = L(x / y) L(x / y) - (1 1 / -1 1) (x / y) = (0 / 0) ({L - 1}...

Homework Statement Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A1>, |A2>, and |A3> such that A|Ak> = ak|Ak>. The system is at state: |psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3>. Provide the possible measured values of a and...

Homework Statement Consider the matrix [1,-5,5;-3,-1,3;1,-2,2] Do four interations of the power method, beginning at [1,1,1] to approximate the dominant eigenvalues of A Homework Equations Matrix multiplication The Attempt at a Solution Okay my issue with this problem is this I...

Homework Statement Let T: M22 -> M22 be defined by T \[ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\] = \[ \left( \begin{array}{cc} 2c & a+c \\ b-2c & d \\ \end{array} \right)\] Find the eigenvectors of T The Attempt at a Solution My...

If C = A +B where A,B are both p.d, than C is p.d and its eigenvalues are positive. Waht can you say about the relationship between the eigenvalues of C, and A,B ? Thanks.

1. The problem statement For integers m >= n, Prove det(xIm - AB) = xm-ndet(xIn - BA) for any x in R. Homework Equations A is an m x n matrix B is an n x m matrix The Attempt at a Solution I tried working out the characteristic polynomials by hand but it just seems too tedious...

Homework Statement Same problem as this old post https://www.physicsforums.com/showthread.php?t=188714 What I'm having problems with is determining the H_{ij} components of the Hamiltonian of a one dimension N site spin chain. And then getting out somehow energy value to prove...

\left( \begin{array}{ccc} 3 - \lambda & 1 & -1 \\ -4 & 2 - \lambda & 2 \\ -2 & 2 & 2 - \lambda \end{array} \right) (3 - \lambda) \left| \begin{array}{cc} 2 - \lambda & 2 \\ 2 & 2 - \lambda \end{array} \right| + 4 \left| \begin{array}{cc} 1 & -1 \\ 2 & 2 - \lambda \end{array}...

Homework Statement Show that two similar matrices A and B share the same determinants, WITHOUT using determinants 2. The attempt at a solution A previous part of this problem not listed was to show they have the same rank, which I was able to do without determinants. The problem is I...

Hi! I don't know much about QM. I'm reading lecture notes at the moment. Angular momentum is discussed. The ladder operators for the angular-momentum z-component are defined, it is shown that <L_z>^2 <= <L^2>, so the z component of angular momentum is bounded by the absolute value of angular...

Prove that a normal operator with real eigenvalues is self-adjoint Seems like a simple proof, but I can't seem to get it. My attempt: We know that a normal operator can be diagonalized, and has a complete orthonormal set of eigenvectors. Let A be normal. Then A= UDU* for some...

Hi there, I have some questions to ask about the topic eigenvalues and one on diagonal matrices 1- can a square matrix exist without eignvalues? Do there exists square matrix without eigenvectors corresponding to each of its eignvalues? 2- What is diagonalisation of a matrix, were abouts...

Homework Statement Proof: \lambda_{\max}(A+B) \leq \lambda_{\max}(A) + \lambda_{\max}(B) Homework Equations Hint from exercise: \lambda_{\max}(A)=\max_{\|x\|=1} x^*Ax The Attempt at a Solution The problem is that the equation on the left side can not be split. So I tried to...

Hi everyone Homework Statement Consider the following 4 x 4 matrix: A = [[6,3,-8,-4],[0,10,6,7],[0,0,6,-3],[0,0,0,6]] Find the eigenvalues of the matrix and their multiplicities. Give your answer as a set of pairs: {[lambda1,multiplicity1],[lambda2,multiplicity2],...} 2...

It seems to me that http://en.wikipedia.org/wiki/Schur_decomposition" relies on the fact that every linear operator must have at least one eigenvalue...but how do we know this is true? I have yet to find a linear operator without eigenvalues, so I believe every linear operator does have at...

Homework Statement x1(t) and x2(t) are functions of t which are solutions of the system of differential equations x(dot)1 = 4x1 + 3x2 x(dot)2 = -6x1 - 5x2 Express x1(t) and x2(t) in terms of the exponential function, given that x1(0) = 1 and x2(0) = 0 The Attempt at a Solution I've already...

Consider the nXn matrix A whose elements are given by, A_{ij} = 1 if i=j+1 or i=j-1 or i=1,j=n or i=n,j=1 = 0 otherwise What are the eigenvalues and normalized eigenvectors of A??

[b]1. What dimensions of a matrix will give repeated complex Eigenvalues? Give an example of one and show that it has repeated complex Eigenvalues. [b]2. No really equations needed? The Attempt at a Solution My attempt is a 2x2 which i don't think is right but here it is. If...

Apparently, if I have a Hamiltonian that contains an operator, and that operator commutes with the Hamiltonian, not only can we "simultaneously diagonalize" the Hamiltonian and the operator, but I can go through the Hamiltonian and replace the operator with its eigenvalue everywhere I see it...

Homework Statement True/False If Ttheta is a rotation of the Euclidean plane R2 counterclockwise through an angle theta, then T can be represented by an orthogonal matrix P whose eigenvalues are lambda1 = 1 and lambda2 = -1. Homework Equations The Attempt at a Solution Just checking to see...

Homework Statement Find the eigenvalues of the following matrix: \left( \begin{array}{ccc} 1 & 0 & -3 \\ 1 & 2 & 1 \\ -3 & 0 & 1 \end{array} \right) Homework Equations The Attempt at a Solution I think I'm forgetting a basic algebra rule or something. I know there are supposed to be 3...

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??

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...