Eigenvectors Definition and 462 Threads

Homework Statement Let v be a non-zero (column) vector in Rn. (a) Find an explicit formula for the matrix Pv corresponding to the projection of Rn to the orthogonal complement of the one-dimensional subspace spanned by v. (b) What are the eigenvalues and eigenvectors of Pv? Compute the...

Homework Statement Write the eigenvector of \sigmax with +1 eigenvalue as a linear combination of the eigenvectors of M. Homework Equations \sigmax = (0,1),(1,0) (these are the columns) The Attempt at a Solution ... Don't know what to do. Can someone show me how to do this using...

Homework Statement Find the principal stresses and the orientation for the principal axis of stress for the following cases of plane stress. σx = 4,000 psi σy = 0 psi τxy = 8,000 psi Homework Equations See picture. The Attempt at a Solution...

Is it true that an nxn symmetric matrix has n linearly independent eigenvectors even for non-distinct eigenvalues? How can we show it rigorously? Basically, I want to prove that if an nxn symmetric matrix has eigenvalue 0 with multiplicity k, then its rank is (n - k). If we can prove that there...

Homework Statement A = \left( \begin{array}{ccc} 2 & 0 & -1 \\ 4 & 1 & -4 \\ 2 & 0 & -1 \end{array} \right) Find the eigenvalues and corresponding eigenvectors that form a basis over R3 Homework Equations The Attempt at a Solution OK so I've found the characteristic...

Let M be a transformation matrix. C is the matrix which diagonalizes M. I'm trying to use the formula D = C-1MC. I noticed that depending on how I arrange my vectors in C, I can change the sign of the determinant. If I calculate D using a configuration of C that gives me a negative value for...

Hi, I'm currently self-teaching myself some mathematics needed to study physics. I'm working through the book Mathematical Methods in the Physical Sciences by Mary L Boas. The book is a well known one, and it's used in many physics programs to teach their math courses. However, I've read the...

he eigenvalues of the 3x3 matrix [[1,-1,-1],[-1,1,-1],[-1,-1,1]] are 2,2, and -1. how can i compute the eigenvectors? for the case lambda=2, for example, i end up with an augmented matrix [[-1,-1,-1,0],[-1,-1,-1,0],[-1,-1,-1,0]] so I'm stuck at this point. much appreciated.

Hi this is my first time posting on here so hopefully I get it right. Given the linear system x'(t) = Ax(t)' with an eigenvalue (lambda) of algebraic multiplicity 2 and geometric multiplicity 1 (repeated root), one solution is w.exp(lambda t) and the other w.t.exp(lambda t) + u.exp(lambda t)...

Homework Statement Use the power method to calculate the dominant eigenvalue and its corresponding eigenvectors for the matrices. The questions are attached with this thread. I have attempted both and seem to have done the first question correctly. I am attempting the second question and am...

Ok, this is starting to come back to me, but I'm stuck again Homework Statement M=\begin{bmatrix} (1-\frac{4}{3}) & 0 \\ -\frac{c}{3} & -c \\ \end{bmatrix} Find eigenvectors and eigenvalues. Homework Equations The Attempt at a Solution Eigenvalues are λ_1=...

I thought I would ask this in the homework section. Homework Statement I should be able to write down the eigenvectors and eigenvalues of diagonal and triangular matrices on sight. M = \begin{bmatrix} 1 &0 \\[0.3em] 0 & x \\[0.3em]...

Another question with respect to finding eigenvectors. If I remember correctly, I should be able to look at certain 2 by 2 matrices and practically write down the eigenvalues and eigenvectors. For example, I have a diagonal matrix, I know immediately what the eigenvalues and eigenvectors...

Ok everybody, it's been awhile since I've taken linear algebra. I need some help dusting off the cobwebs. (I'm trying to follow this in a paper; this isn't a homework question, but I'll be glad to move it...) Let's say I have a matrix M = \begin{bmatrix} -σ & σ & 0...

http://dl.dropbox.com/u/33103477/Untitled.png My solution: M=\begin{bmatrix} t+15 & -12 \\ 24 & t-19 \end{bmatrix} The eigen values are 1,3. Hence as the matrix has real and distinct eigenvalues it is diagonalisable. Now the characteristic equation is t^2 - 4t +3 =0...

How do I determine powers of matrices(2x2) without calculating their eigenvectors and doing the pdp^-1 thing ? Obviously multiplying over and over is not a solution.

I've been having some trouble with conceptually understanding the idea of a generalized eigenvector. If we have a linear operator A and want to diagonalize we get it's eigenvalues and eigenvectors but if the algebraic multiplicity of one of the eigenvalues is greater than the geometric...

Homework Statement I'm having a problem with a question. I need to find the transition matrix in the form T=UAU^-1 where U=[V1 V2] Homework Equations T=UAU^-1 where U=[V1 V2] The Attempt at a Solution my original transition matrix is [0.9 0.002; 0.1 0.998] from that i calculated...

Too many Eigenvectors!? Homework Statement I have to find the eigenvalues and eigenvectors of: -1 2 -2 1 2 1 -1 -1 0 and I can find four eigenvectors I'm not sure how to tell which of my eigenvectors is wrong as they all seem to satisfy Av=λv (i also checked that they arent...

Hi! I am a new user who is not an expert with Linear Algebra at all. I have some questions about eigen values/vectors and their meaning with relation to a 2x2 matrix, or tensor, which was obtained by the tensor product of 2 vectors. First, I have two 2-dimensional 2x1 vectors "v1" and "v2"...

This seems a simple question but I can't find the solution by myself. Please help. Say we have a 2 by 2 matrix with different eigenvalues. Corresponding to each eigenvalue, there are a number of eigenvectors. Q1. Could the eigenvectors corresponding to the same eigenvalue have different...

Homework Statement One of the problems in our test is this. Express the eigenvectors of J_y in terms of the eigenkets of J^2 and J_z . Homework Equations The Attempt at a Solution I know the matrix of J_y and the operators or eigenkets for J_y , J^2 and J_z . I just don't seem...

Homework Statement The Attempt at a Solution So, first I wrote, T(X) = λ_1 X, T(Y) = λ_2 Y If λ_1 = λ_2: T(X+Y) = T(X) + T(Y) = λ_1 X + λ_2 Y = λ_1 (X+Y), so this does indeed seem to be an eigenvector. But I'm less convinced for the case λ_1 ≠ λ_2. Again, I get the...

For spin-1/2, the eigenvalues of S_x, S_y and S_z are always \pm \frac{\hbar}{2} for spin-up and spin-down, correct? What is the difference between eigenvectors, eigenstates and eigenspinors? I believe eigenstates = eigenspinors and eigenvectors are something else? I'm just getting confused...

Find the eigenvalues and corresponding eigenvector of the matrix. A= [-4 4 8 ] [0 0 -10] [0 0 2 ] [1 -1 0] ~ [0 0 1 ] [0 0 0 ] I calculated by A = -\lambdaI So, [1-lamda -1 0 ] [0 -lamda 1] [0 0 -lamda] so, lamda = 0,0, and 1 So I got...

Hey guys, I need to find the equilibrium solution (critical point) for the given system. Also I need to take the homogeneous equation x' = Ax (matrix notation) and find the eigenvalues and eigenvectors. system: x' = -x - 4y - 4 y' = x - y - 6 Can you help? Thanks

One more question please... which one of these statements is NOT true (only one can be false): a. a square matrix nXn with n different eigenvalues can become diagonal. b. A matrix that can be diagonal is irreversible. c. Eigenvectors that correspond to different eigenvalues are linearly...

Homework Statement The Attempt at a Solution T(1,0,0) = (3,-1,0) T(0,1,0) = (0,1,0) T(0,0,1) = (-1,2,4) Thus, we have the matrix, \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right| Δ_T (t) = det( \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right|...

We are aware that by knowing the eigenvalues and eigenvectors we can evaluate the determinant, say if it is invertible and diagonalize to find powers of matrices. Is there a list of properites of a matrix we can find by eigenvalues and eigenvectors? Are there things that e.values and e.vectors...

well, if a matrix has n linearly independent eigen-vectors then it's easy, what if a matrix is not diagnolizable in that way? Can we still diagnolize it by other means? And what if a matrix is not diagonalizable at all? Are there still ways to find its exponential matrix?

M = (a c) (c b) Sorry for the double sets of brackets, its all in one. I'll also show as far as i got below: [a-λ c] => (a-λ)(b-λ) - c^2 = λ^2 + (-a-b)λ + (ab-c^2) =0 [c b-λ] => then using the quadratic formula: λ = [-(-a-b) +/- Sqrt{(-a-b)^2 - 4(1)(ab-c^2)}]/ 2 then...

Homework Statement Calculate the eigenvectors and eigenvalues of the two-dimensional matrix representation of the Hermitean operator \hat{O} given by |v_k'>\left(O|v_k>= {{O_11,O_12},{O_21,O_22}} where all Oij are real. What does Hermiticity imply for the o- diagonal elements O12...

Hello, sorry that I am asking too many questions, I am preparing for an exam... I have a matrix, 0 1 0 0 0 0 0 0 1 and I need to say if it has a diagonal form (I mean, if there are P and D such that D=P^-1*D*P) I found that the eigenvalues are 0 and 1. I also know that if I use 0, I get the...

I am little confused about the choice of eigenvectors chosen by my book. I am wondering if an eigenvalue can have multiple eigenvectors and if all are equally correct. Case in point the example below: Homework Statement find a fundamental matrix for the system x'(t) = Ax(t) for the given...

Homework Statement Calculate the eigenvalues and eigenvectors of the operator, J.n, where n is a unit vector characterized by the polar angles theta and phi, and J is the spin-1 angular momentum operator. Homework Equations Matrix representations for J^2 and J(z) The Attempt at a...

Homework Statement Calculate the Eigenvalues and eigenvectors of H= 1/2 h Ω ( ]0><1[ + ]1><0[ ) Homework Equations I know H]λ> = λ]λ> The Attempt at a Solution I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?

I have no trouble calculating eigenvalues but I have a hard time understanding how to use them. I know that you can somehow calculate a bridge's self-frequency with eigenvalues but I don't know how. What I am after is, what do eigenvectors and eigenvectors mean physically or in other ways? I...

"Completeness" of eigenvectors in a complete, commuting set Hi guys, I asked this question on Physics Stack Exchange many days ago and even after substantial discussions and revisions, this has remain unanswered. This is...

My aim was to numerically calculate eigenvalues and eigenvectors for a square A matrix. I managed to find the eigenvalues by using QR algorithm. Now, I can find all of the eigenvalues for any given square matrix. But, for the next step, how do I find the corresponding eigenvectors? Is there...

Homework Statement In my quantum class we learned that if two operators commute, we can always find a set of simultaneous eigenvectors for both operators. I'm having trouble proving this for the case of degenerate eigenvalues.Homework Equations Commutator: [A,B]=AB-BA Eigenvalue equation:A...

Let A =\begin{bmatrix} -14 & 5 & -2 & 1 \\ -38 & 13 & -4 & 5 \\ 25 & -10 & 7 & 5 \\ -17 & 5 & -2 & 4 \end{bmatrix} . The Jordan Normal Form of A is J =\begin{bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 3 \end{bmatrix} . Now, I've been told that eigenvectors...

In the last 2 weeks we've begun learning about eigenvalues/vectors. It will come up in my exam in January so I'm trying hard to get my head around this. I've tried various different sources to learn this but I'm hoping someone here can offer a different view on it. Basically, I can work out the...

I got a 5x5 matrix, if use characteristic equation to find the eigenvalues and eigenvectors are very tedious and trouble, so got any method which are easy to calculate?

Homework Statement The Markov matrix A = [.9 .3; .1 .7] has eigenvalues 1 and .6, and the power method uk=Aku0 converges to [.75 .25]T. Find the eigenvectors of A-1. What does the inverse power method u-k=A-1u0 converge to (after you multiply by .6k)? Homework Equations The...

Homework Statement Convert y"=0 to a first-order system du/dt=Au d/dt [y y']T = [y' 0]T = [0 1; 0 0] [y y']T This 2x2 matrix A has only one eigenvector and cannot be diagonalized. Compute eAt from the series I+At+... and write the solution eAtu(0) starting from y(0)=3, y'(0)=4. Check...

Please note: Below, I keep trying to put [ capital B ] but it gets turned into [b]! In Dennery and Krzywicki, they give an example of how to put a matrix in Jordan canonical form (pp. 167-170). They start with a 4x4 matrix [A] that looks kind of messy and transform it to a quasi-diagonal form...

Homework Statement Use eigenvalues and eigenvectors to find the general solution of the system of ODEs.. x1 = 3x1 - x2 x2 = -x1 + 2x2 - x3 x3 = -x2 + 3x3 Homework Equations The Attempt at a Solution I converted that into the matrix...

I have to be able to figure out eigenvalues and eigenvectors for 2x2 and 3x3 matrices for my physics course, but I have never taken linear algebra so I obviously have no idea what they even are. I need someone to basically teach me how to solve these problems because I have no knowledge of this...

Homework Statement Let B be a matrix with characteristic polynomial λ2-λ√6+3. Evaluate B4. Homework Equations Bn=PDnP-1 The Attempt at a Solution I can find the eigenvalues from the characteristic equation and those would form the diagonal entries of D. But how would I find P, which contains...

Hello This is a concept question I do not understand. I'm just wondering why the answer is what it is. (the answer is written below the question, I just have no idea where it comes from)