Cleve's Corner

eigshow, week 2

Three more examples with eigshow, all of them degenerate in some way or another.

Contents

A scalar matrix

The first example is a scalar multiple of the identity matrix. It is not available from the pull-down menu in the eigshow title, but eigshow can be called with a 2-by-2 matrix argument.

format rat, format compact
A = [5 0; 0 5]/4

A =
5/4            0
0              5/4


The blue ellipse is a circle. Ax is always equal to (5/4)*x and so A*x is always parallel to x. Any vector is an eigenvector. 5/4 is a double eigenvalue. eigshow chooses two unit vectors as the eigenvectors but it could have chosen any two vectors.

A singular matrix

A = [2 4; 2 4]/4

A =
1/2            1
1/2            1


The blue ellipse has collapsed to a line. It may be a hard to see when A*x is parallel to x, but it is when x itself is in the line, or when A*x is the zero vector. The corresponding eigenvalues are 3/2 (the half-length of the line) and zero. [Note added Aug. 5, 2013: This is not correct. See my comment posted today.]

A defective matrix

A = [6 4; -1 2]/4

A =
3/2            1
-1/4            1/2


Matrices with this type of behavior are rare, but important. The blue ellipse is not degenerate, but as you move around the unit circle Ax lines up parallel for only one x and its negative. This is a matrix of order two with only one eigenvector. It has a double eigenvalue equal to 1 and a nondiagonal Jordan Canonical Form. Such matrices are known as defective and play an important role in the theory of ordinary differential equations and dynamical systems.

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4 Responses to “eigshow, week 2”

1. Andrew replied on :

Very nice exposition. Thank you. I just designed a dynamical system such that its Jacobians would have N-1 repeated (and stable) eigenvalues at particular equilibrium points. Indeed, putting the Jacobians in Jordan canonical form (jordan in Matlab) results in a non-diagonal matrix. I don’t know if I like the idea of calling the perfectly good Jacobians “defective” though … but I’ll get used to it.

2. Cleve Moler replied on :

My parenthetical remark in the paragraph about the singular matrix that the length of the line is the eigenvalue is incorrect. The length is actually the singular value, as pointed out in the forthcoming week 3 post. This matrix is not symmetric, so the eigenvector is not the axis of the degenerate ellipse. The eigenvalue is 3/2 = 1.50; the singular value is sqrt(10)/2 = 1.58.

3. Rich replied on :

For the last example you say it lines up for only one x, but at least visually it really looks like there’s an alignment both in the IVth and IInd quadrants where the circle and ellipse intersect. Why doesn’t this then count as two eigenvectors?

4. Cleve Moler replied on :

This is a good question, Rich. I’m glad you asked. The two vectors you are seeing are negatives of each other, x and -x. If x is an eigenvector, then Ax = lambda*x and A*(-x) = lambda*(-x), so the two vectors really count as the same eigenvector. In fact, any multiple of x is still acts as the same eigenvector.

Cleve Moler is the author of the first MATLAB, one of the founders of MathWorks, and is currently Chief Mathematician at the company. He is the author of two books about MATLAB that are available online. He writes here about MATLAB, scientific computing and interesting mathematics.

These postings are the author's and don't necessarily represent the opinions of MathWorks.