[U,V]=eig(A) , AU won't equal UV

My understanding with [U,V]=eig(A) is that AU=UV, however, in some cases with matlab, these this is not true.

It works with A=diag([3, 3, 3, 2, 2, 1]), but not with A=magic(4) or A=ones(4)

Is AU=UV supposed to hold true or will they not equal each other in certain cases?

Answer 1

[Bhaskar R]

In general terms A*U and U*V are equal but in MATLAB it computes with difference in decimal points almost at 15th decimal value as you see

A=magic(4);
[U,V]=eig(A);
format long

>> A*U
  -17.000000000000000  -7.366563145999501  -3.366563145999497  -0.000000000000001
 -17.000000000000000   3.788854381999832  -0.211145618000169   0.000000000000007
 -17.000000000000000   0.211145618000165  -3.788854381999833   0.000000000000004
 -17.000000000000000   3.366563145999499   7.366563145999495   0.000000000000006

 >> U*V
 -17.000000000000000  -7.366563145999498  -3.366563145999494   0.000000000000000
 -16.999999999999993   3.788854381999834  -0.211145618000169   0.000000000000001
 -17.000000000000004   0.211145618000170  -3.788854381999831  -0.000000000000001
 -16.999999999999996   3.366563145999493   7.366563145999494  -0.000000000000000

as you can see that there is a difference at 15th decimal point thats why your if condition is failing each time

And one more thing you can't use A*U == U*V in if condition cause it produces matrix instead of a logical condition, use as isequal(A*U, U*V).

Answer 2

[Erivelton Gualter]

It is not a bug.

It depends of the floating point numbers. If you type the following, you will observe that the floating point is equivalent to 2.2204e-16:

eps

Now, try the following:

A*U - U*V

It is almost zero. Maybe aroung 10^-15 for your case.

There is a better explanation about this matter. But for sure, I know you do not need to worry about the eig function contains a bug or not.