Eigenvalues and Eigenvectors Question

Question

LukeO 2016-11-18

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评论： Greg Bishop 2017-2-15

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I've solved a matrix in MATLAB using the following in the command window:

>> A = [1 2 -5; 3 7 3; -4 6 1]

A =

     1     2    -5
     3     7     3
    -4     6     1

>> eig(A)

ans =

   -5.4490
    5.6560
    8.7930

>> [v,d] = eig(A)

v =

    0.6404   -0.7188   -0.2483
   -0.3223    0.0583    0.7017
    0.6971    0.6927    0.6677

d =

   -5.4490         0         0
         0    5.6560         0
         0         0    8.7930

I understand the function eig(A) is returning the eigenvalues of the matrix but I don't fully understand the function [v,d]=eig(A). Could someone please explain it in a way that is easy to understand?

Also, I understand there are many different ways of reporting the eigenvector but which eigenvector does MATLAB report?

Finally, is there a way in MATLAB that i can confirm the above answers for the eigenvalues and eigenvectors just to prove its correct?

Thanks.

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Greg Bishop 2017-2-15

https://www.mathworks.com/help/matlab/ref/eig.html

V are the eigenvectors, D are the eigenvalues, the algorithm used to compute them is cholesky which is a triangular decomposition. By definition, you can check this by performing A*V = V*D, within the machine limits of float precision.

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Answer 1

Star Strider 2016-11-18

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According to the documentation, ‘v’ in your example are the right eigenvectors. See the documentation for eig for a full explanation.

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Answer 2

Roger Stafford 2016-11-18

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The ‘eig’ function “[v,d] = eig)A)” is a solution to the equation

A*v = v*d

where ‘d’ is a diagonal matrix and if possible the columns of ‘v’ are of unit magnitude. These diagonal elements of ‘d’ are known as the eigenvalues of A and the columns of ‘v’ are the eigenvectors. The above equation is a way of testing that ‘eig’ has solved the problem correctly (to within round-off error accuracy, of course.) For further information, read:

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

The subject of eigenvalues and eigenvectors is an important subject in linear algebra.