What is Eigenvectors: Definition and 458 Discussions

In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by



λ


{\displaystyle \lambda }
, is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.

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  1. A

    I don't get Eigenvalues or Eigenvectors

    I just finished Differential Equations, and I know how to find eigenvalues/eigenvectors, and I understand how to use them to solve a differential equation. But I don't really understand "what they are". How is a matrix with complex eigenvalues any different than a matrix with real...
  2. T

    Finding Eigenvectors of Matrix A

    Homework Statement I have a matrix A = [3 -2; -3 2], then I peform det(A-λI) and find solutions λ=0 and λ=5. For the first case λ=0 I perform A-0I, which of course is just A [3 -2; -3 2]. I then reduce to row echelon form and find the values V1 = 2 and V2 = 3, therefore for λ=0 my...
  3. J

    Linear Algebra: Diagonalization, Transpose, and Disctinct Eigenvectors.

    Homework Statement Show that if an nxn matrix A has n linearly independent eigenvectors, then so does A^T The Attempt at a Solution Well, I understand the following: (1) A is diagonalizable. (2) A = PDP^-1, where P has columns of the independent eigenvectors (3) A is...
  4. H

    Eigenvalues and Eigenvectors of 2x2 Matrix | Homework Help

    Homework Statement Find the eigenvalues and correspointing eigenvectors of the matrix: [1,1;1,1] Homework Equations The Attempt at a Solution I can solve the determinant to get the eigenvalues: e1=2, e2=0, and from here I am supposed to sub these values back into my matrix...
  5. S

    Linear Algebra - Eigenvalues / Eigenvectors

    I have two problems here; one I think I almost have but I'm stuck, and the other I'm pretty much stumped on.Homework Statement Suppose V is a complex vector space and T is in L(V). Prove that T has an invariant subspace of dimension j for each j = 1, ... dim(V). Homework Equations The Attempt...
  6. T

    A question about a new way to find eigenvectors that i noticed

    i got this question in which we are given the matrix T and we need to find the eigenvalues and the independent spaces (i don't know what is independent space) of T^2 +2*T the problem is that he started to solve the question as i would have solved it but then he puts a big X on it and does...
  7. S

    Lin. Algebra - Find Eigenvectors / eigenvalues

    Homework Statement Define T in L(F3) by T(z1, z2, z3) = (2*z2, 0, 5*z3). Find all eigenvalues and eigenvectors of T.Homework Equations The Attempt at a Solution Well, since we want to find all the eigenvalues, we want the following equation to hold: T(z1, z2, z3) = (2*z2, 0, 5*z3) = \lambda(z1...
  8. A

    Diagonalization, eigenvectors, eigenvalues

    [SOLVED] diagonalization, eigenvectors, eigenvalues Homework Statement Find a nonsingular matrix P such that (P^-1)*A*P is diagonal | 1 2 3 | | 0 1 0 | | 2 1 2 | Homework Equations I am at a loss on how to do this. I've tried finding the eigen values but its getting me...
  9. N

    Eigenvectors and their inverses

    [SOLVED] Eigenvectors and their inverses Homework Statement After submitting the first question, I thought of a new one - so there are two questions: 1) I have a n x n matrix A and it has n (not necessarily different) eigenvalues. I can write the matrix A as the product of: S*D*S^(-1)...
  10. B

    EigenValues & EigenVectors proofs

    Question 1: Proove that if λ is an eigenvalue of [A], then 1/λ is an eigenvalue of [A]{T} Question 2 Proove that a square matrices [A] and [A]T have the same Eigenvalues. Question 3: Show that |det(A)| is the product of the absolute values of the eigenvalues of [A]...
  11. I

    Finding Eigenvectors & Eigenvalues of A Matrix

    Homework Statement Use direct multiplication to show that for each of the following matrices A, the given vectors v1, v2, and v3 are eigenvectors of A and to find the eigen values lama1, lama2, and lama3 of A: A=top row: (2 -1 3) second row: (-1 6 -1) third row: (3 -5 2)...
  12. M

    Practical applications of eigenvectors

    Homework Statement A simple eigenvector problem using a 2x2 matrix. Homework Equations Ax = \lambda x The Attempt at a Solution I don't have one. I know and understand the theory behind eigenvectors, but I cannot think of a practical application. I need to create a problem that uses...
  13. K

    How do I find the eigenvalues and eigenvectors of a 3x3 matrix A?

    Homework Statement Find the eigenvalues and eigenvectors of A 1 1 a 1 -1 b 0 0 1 you can assume a and b are not equal to zero Homework Equations The Attempt at a Solution Using det(A-sI) = 0 where s is the eigenvalue i get (s^2 - 2)(1 - s) = 0 therefore giving...
  14. C

    Proving kernel of matrix is isomorphic to 0 eigenvalue's eigenvectors

    Homework Statement I want to prove that the eigenvectors corresponding to the 0 eigenvalue of hte matrix is the same thing as the kernel of the matrix. Homework Equations A = matrix. L = lambda (eigenvalues) Ax=Lx The Attempt at a Solution Ax = 0 is the nullspace. Ax = Lx...
  15. B

    Eigenvectors of a 3x3 Matrix

    Homework Statement Given Matrix B: [ 1 2 1] [-1 2 -1] [ 2 -2 3] and knowing that one of the Eigenvalues is 4, find one other value and its corresponding eigenvector Homework Equations Bx=Lx (The basic idea behind eigenvectors) det(B-LI)=0 The Attempt at a Solution Ive...
  16. E

    Eigenvectors, eigenvalues and matrices

    I have: x' = \left(\begin{array}{cc}2&-5\\1&-2\end{array}\right) x I found that the eigenvalues are r_1 = i and r_2 = - i. Also, I calculated the eigenvectors to be \xi_1 = \left(\begin{array}{c}2 + i\\1\end{array}\right) \xi_2 = \left(\begin{array}{c}2 - i\\1\end{array}\right)...
  17. R

    Real Eigenvalues and 3 Orthogonal Eigenvectors for Matrix (c,d)

    Homework Statement For which real numbers c and d does the matrix have real eigenvalues and three orthogonal eigenvectors? 120 2dc 053 Homework Equations im having trouble getting started on this one. Ive tried using solving for the eigenvalues pretending that c and d are...
  18. N

    Eigenvalues and eigenvectors

    Hi I came across a problem of eigenvalues and eigenvectors. It was easy and I solved it but one thing made me unsure about the answer. All the three eigenvectors were zero vectors. Here is the question and my answer: The matrix A= ( -1 0 0 1 0 -2 0 0 0 1 -2 0...
  19. Q

    Bases of Generalized Eigenvectors

    Hi all. So I'm a bit confused about finding a basis of generalized eigenvectors for an operator that is not diagonalizable. I have some "steps" in mind, but maybe someone can help me out here: 1) Find the eigenvalues of the matrix/operator 2) Find the eigenspaces corresponding to each...
  20. S

    Eigenvalues and eigenvectors of a linear transform

    Homework Statement Find all the eigenvalues and eigenvectors of the linear transformation: T(f) = 5f ' -3f T: from C^(nfnty) --> C^(nfnty) where C^(nfnty) is set of continuously functions Homework Equations A scalar B is called an eigenvalue of T if there exists a nonzero element f...
  21. K

    Eigenvalues And Eigenvectors Problems

    1. How to show (prove) the Cayley-Hamilton theorem : “Every matrix is a zero of its characteristic polynomial , Pa(A)=0”. 2. A and B are n-square matrices, show that AB and BA have the same eigenvalues. 3. Show that to say that “ 0is an eigenvalue of linear mapping U” is equivalent to “ U...
  22. A

    Solving eigenvectors of Operator (a+)^2-(a)^2

    Homework Statement I need to find the eigenvectors of the following operator (a+)^2-(a)^2, when (a+), (a) are the creation and the annihilation operators. Homework Equations The Attempt at a Solution I tried to put the eigenvectors as sum of eigenvectors of operator...
  23. J

    Finding eigenvectors of a matrix

    Homework Statement I need to find the eigenvectors for the matrix shown below. Homework Equations Nothing relevant. The Attempt at a Solution I have the matrix 2 1 -1 0 0 4 -2 0 0 3 -1 0 0 3 -2 1 The eigenvectors are 1,1,2,2. To get the eigenvectors for 1,1, I use the...
  24. K

    Eigenvectors & normalising

    Homework Statement Hi, I'm going over an old exam paper as part of my revision for upcoming exams (joy ! ok, maybe not :yuck: ) Anyway, I've gotten myself a bit lost here and would appreciate some guidance. Here we go: Compute the eigenvalues & eigenvectors of the following matrix...
  25. A

    Eigenvalues and Eigenvectors

    Suppose that B is the inverse of A. Show that if |psi> is an eigenvector of A with eigenvalue a not equal to 0, then |psi> is an eigenvector of B with eigenvalue 1/a. So I know that A|psi> = a|psi>, and I'm trying to prove that A^(-1)|psi> = 1/a|psi>. I tried simplifying A as a 2x2 matrix...
  26. T

    How to Solve Complex Eigenvectors in Matrix Algebra

    Hey, I have a quick question that I can not seem to find much of an answer to in my text. When working with a nxn matrix, A, and you find eigenvalues that are complex, I'm confused about how to go about finding the actual eigenvector. I know we compute the null space of A-lambdaI, but that is...
  27. N

    Finding Eigenvectors by inspection

    Would someone please explain to me how I can find eigenvalues and eigenvectors by inpection of simple symmetric matrices? I just can't figure it out. He is an example: By looking at A=\left(\begin{matrix}2&-1&-1\\-1&2&-1\\-1&-1&2\end{matrix}\right) I should be able to guess...
  28. G

    Finding Quantum Numbers and Eigenvectors from Matrices A & B

    Hey, I have two matrices A and B which commute. For A I have 1,-1,-1 and for B I have 1,2,2. I am asked to find the quamtum number for the three states. How to find the quantum states from the eigenvalues. It is further said that it is possible to find the eigenvectors from the quantum...
  29. Z

    What are the eigenvectors |E1> and |E2> of the energy?

    |a'> and |a"> are both eigenvectors of eigenvalue a' and a" of an Operator A and a' doesn't equal a",The hameltonien of the system is defined as H=ε( |a'><a"| + |a"><a'| ) a)What are the eigenvectors |E1> and |E2> of the energy? b)If the system was in the state |a'> at t=0 , write the system...
  30. K

    Help with the concept of eigenvectors

    Hi, I just need a little bit of help with the concept of eigenvectors. I have a basic 2x2 matrix and have found the eigenvalues to be: 4 and 9 I have also tried going through the process of finding the eigenvectors that my lecturer has shown me, but I'm not sure where to go from there, I'm...
  31. I

    Issue regarding the orthogonality of eigenvectors for Hermitian

    At the risk of arrousing the ire of the moderaters for posting the same topic in two forums, I again ask this question as no one in the quantum forum seems to be able to help. So... Regarding a proof of the orthogonality of eigenvectors corresponding to distinct eigenvalues of some Hermitian...
  32. I

    Problem with proof of orthogonality of eigenvectors for Hermitian

    I'm not sure if this is the appropriate section, perhaps my question is better suited for Linear Algebra. At any rate, here goes. Regarding a proof of the orthogonality of eigenvectors corresponding to distinct eigenvalues of some Hermitian operator A: Given A|\phi_1\rangle = a_1|\phi_1\rangle...
  33. T

    Finding eigenvalues and eigenvectors of a matrix

    I'm asked to find the eigenvalues and eigenvectors of an nxn matrix. Up until now I thought eigenvectors and eigenvalues are something that's related to linear transformations. The said matrix is not one of any linear transformation. What do I do?
  34. S

    How can I correctly find the eigenvectors for this matrix?

    find the eigenvectors for this matrix \left( \begin{array}{ccc}3&0&6\\0&-3&0\\5&0&2 \end{array} \right) easy to find the eigenvectors which are -3,-3 and 8 now how to find the eigenvectors am i supposed to do \left| \lambda I - A \right| X = 3 X ? then \left( \begin{array}{ccc}...
  35. fargoth

    Finding eigenvalues and eigenvectors

    is there any trick for finding the eigenvalues and vectors for this kind of matrix? \left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & \sqrt{\frac{3}{2} & 0 & 0 \\ 0 & \sqrt{\frac{3}{2} & 0 & \sqrt{\frac{3}{2} & 0 \\ 0 & 0 & \sqrt{\frac{3}{2} & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array}...
  36. K

    Why Do Eigenvectors Confuse Me So Much?

    Eigenvectors + Me= ?AHHHHHHHHHHH I am really trying to understand Eigenvectors but you have to understand that my prof. only teaches about HOW to get the eigenvalues/vectors and how to use then to solve diff. eqs. So far I can find the eigenvalues of a 2x2 and from there I can get the...
  37. W

    Finding eigenvectors and eigenspinors of S_y

    I have a homework problem whose first part asks for the eigenvectors and eigenspinors of S_y . My problem is following the text to figure out what (mathematically) is S_y . Also, The book first derives S^2, using the following method using 3/4\hbar^2 as the eigenvalue. I was orignally...
  38. J

    What is the correct eigenvector for lambda=4 in [8 -10;2 -1]?

    I am studying for a test that I have today and just need a quick anwser. the problem is find the eigenvector of [8 -10;2 -1] where ; means skip to a new line. So I get lamda=4,3 then try to find eigenvector for lamda=4. I get the vector [5/2;1] which works but the anwser in the book is [2;5]...
  39. M

    Eigenvectors, Eigenvalues and Idempotent

    I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help. Question: An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must...
  40. F

    Where am I going wrong with complex eigenvectors?

    I'm having trouble with this problem. Actually I'm having trouble with all of this set of problems (when the eigenvectors are complex). I must not be finding these things correctly, because nothing is matching up with the book. Any help would be awesome. \vec {x}\,' = \left( \begin{array}{cc}...
  41. B

    Finding Eignevalues & Eigenvectors of A⁻¹ Without Direct Computation

    Hi, I need help for this problem: Find the eignevalues and eingenvectors for the matrix below. DO NOT compute them directly by computing the matrix: A-1 We need to find some kind of demonstration to see if the eignevalues of A-1 are the same, opposite or inverse (or whatever) as those of...
  42. M

    Confused on finding Eigenvalues and Eigenvectors

    confused on finding Eigenvalues and Eigenvectors! hello everyone, i can't understand this example, how did they find the Eigen value of 3?! Aslo an Eigen vector of 1 1? http://img438.imageshack.us/img438/1466/lastscan1oc.jpg thanks.
  43. O

    What are eigenvalues and eigenvectors?

    hi, i have trouble understanding these two terms. can anyone explain to me eigenvalues and eigenvectors in laymen terms? Thks in advance! :smile:
  44. P

    Finding Eigenvectors for Coordinate transformation

    I'm wondering if anyone here might have a solution to a problem I've having. This is a Quantum Mechanics problem I'm doing. I calculate a 4 by 4 complex Hermitian matrix (H = Hamiltonian) in a basis where it is not diagonal. I diagonalize it numerically (using eispack) and get eigenvalues...
  45. J

    Eigenvalues, eigenvectors question

    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...
  46. M

    Understanding Eigenvectors: Solving for Eigenvectors of a 2x2 Matrix

    The eigenvalues of the matrix \left(\begin{array}{cc}0 & \frac{1}{2}\\\frac{1}{2} & 0\end{array}\right) are \lambda_1 = \frac{1}{2} and \lambda_2 = -\frac{1}{2} The problem here is that I have no idea of how to calculate the eigenvectors. Could some one please explain me, in detail, how...
  47. M

    How do I find the eigenvectors for this matrix?

    The eigenvalues of the matrix \left(\begin{array}{cc}0 & \frac{1}{2}\\ \frac{1}{2} & 0\end{array}\right) are \lambda_1 = \frac{1}{2} and \lambda_2 = -\frac{1}{2} The problem here is that I have no idea of how to calculate the eigenvectors. Could some one please explain me, in detail, how...
  48. Z

    Eigenvectors and using them in matrix algebra.

    Hi. Marix A= |1 1 0 | |0 2 0 | |2 1-1 | Has three eigenvectors [1,1,1]^T, [1,0,1]^T and [0,0,1]^T, By using this knowledge solve A^11. Ok, solving A^11 is rather easy with any decent calculator, or even with pen , paper and some time, but how on Earth I'm supposed to benefit...
  49. M

    Hairy billiard balls and eigenvectors

    this is no doubt older than dirt, but it just popped into my head when trying to think of how to convince my beginning linear algebra class that every linear isometry of R^3 has an eigenvector: i.e. it follows from the theorem that one cannot comb the hair on a billiard ball. do you see how...
  50. D

    Diagonalize a Matrix A - Normalize eigenvectors

    I have a homework problem here I am a little at a loss on due to not very good examples in class and the part of the book that explains them is 4 chapters ahead and loaded with words I just do not understand yet. :bugeye: If someone could give a definition or two and get me started on this bad...
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