Eigenvectors not quite right

[anthonyQueen]

I have the following puzzle, that I can illustrate with this simple example, but that I fear it is more general

With a 2x2 matrix like this:

A:=

0.2	1
0	0.3

I applied the simple command "eigenvecs(A), with the result of:

1	0.995
0	0.1

Now, the second eigenvector does not seem precisely right. It should be 1, 0.1, and not 0.995; 0.1.

I tested this with two different versions of Mathcad, with the same result. Can someone kindly explain if it is my mind wrong, or if it is the application that somwhat fails in precision?

In the latter case, is there something that can be done to make Matchad more precise, in so simple calculation?

Thanks in advance.

[AlanStevens]

Mathcad's eigenvectors are normalised to unity.

Alan

[anthonyQueen]

Many thanks Alan for the answer. I did not get it completely, though.

Does "normalized to unity" mean that the maximum value is 1? The answer I proposed did not violate that.

In any case, whatever kind of normalization is applied (and I am curious to know which one), the ratio of the two elements of the second eigenvector should be equal to ten. So either (transpose): [1; 0.1] or [0.995; 0.0995] or any other combination which ratio is exactly 10.

So I guess that probably there is an approximation from 0.0995 to --> 0.1. Is there any way to change the default approximation on screen? I am usually on Mathcad v11. Thanks again.

[AlanStevens]

anthony Queen wrote:

Many thanks Alan for the answer. I did not get it completely, though.

Does "normalized to unity" mean that the maximum value is 1? The answer I proposed did not violate that.

Normalised to 1 means that,, if the two components are, say a and b, then sqrt(a^2+b^2) = 1.

Alan

[anthonyQueen]

Thanks Alan. I also find how to increase the number of decimal places, so that the number "0.1" is left as 0.0995 (Menu Format / Result...).

One last thing I see still puzzling. The command "eigenvec(M,z)" returns the eigenvector of M associated with the corrispondent eigenvalue, z. Yet in my case it adds negative signs that are not present in the solution of my first post.

To be clear (see first post) : in the matrix A the value of z are z_0=0.2 and z_1=0.3.

But the result of eigenvec(A, z_0) is [-1; 0] which shows a negative sign not present in the eigenvectors matrix derived with "eigenvects(A)".

That sounds strange: should not the two results be the same?

[AlanStevens · Accepted solution]

anthony Queen wrote:

One last thing I see still puzzling. The command "eigenvec(M,z)" returns the eigenvector of M associated with the corrispondent eigenvalue, z. Yet in my case it adds negative signs that are not present in the solution of my first post.

To be clear (see first post) : in the matrix A the value of z are z_0=0.2 and z_1=0.3.

But the result of eigenvec(A, z_0) is [-1; 0] which shows a negative sign not present in the eigenvectors matrix derived with "eigenvects(A)".

That sounds strange: should not the two results be the same?

No. The following is from the help file:

• The results returned by eigenvec and eigenvecs are not necessarily identical. For a given eigenvalue, there are infinitely many eigenvectors, and the one found depends on the algorithm used. Every eigenvector for a particular eigenvalue is a multiple of other eigenvectors.

Alan