MaplePrimes - answers and comments on Question, error in JordanForm

repeated roots

PatrickT — Thu, 18 Mar 2010 21:54:15 Z

Here (http://en.wikipedia.org/wiki/Diagonalizable_matrix) it states:

An n-by-n matrix A is diagonalizable over the field F if it has n distinct eigenvalues in F, i.e. if its characteristic polynomial has n distinct roots in F.

restart:
A := Matrix([[1,sqrt(2),-sqrt(6)],[sqrt(2),2,sqrt(3)],[-sqrt(6),sqrt(3),0]]);

                         [          1/2      1/2]
                         [  1      2       -6   ]
                         [                      ]
                    A := [ 1/2              1/2 ]
                         [2         2      3    ]
                         [                      ]
                         [  1/2     1/2         ]
                         [-6       3         0  ]

LinearAlgebra:-JordanForm(A);
LinearAlgebra:-CharacteristicPolynomial(A,x);

                3      2                 1/2  1/2  1/2
               x  - 3 x  - 9 x + 15 + 2 6    2    3

solve(%,x);

                               -3, 3, 3

In other words, the root 3 is repeated, so -- if the theorem stated in wikipedia is correct -- the matrix is not (necessarily) diagonalizable.

Bug

Robert Israel — Thu, 18 Mar 2010 23:42:12 Z

The problem is apparently one involving simplification: at some point Maple doesn't recognize that sqrt(6)=sqrt(3)*sqrt(2). If you replace sqrt(6) by sqrt(2)*sqrt(3), you get the correct Jordan form. Note also:

> Q:= JordanForm(A, output='Q');

which is the Matrix that should produce Q^(-1) . A . Q = J. But in fact, this Q is singular: its second column simplifies to all 0.

> simplify(Q);
                   [       1                    2        ]
                   [       -         0          -        ]
                   [       3                    3        ]
                   [                                     ]
                   [    1  (1/2)            1  (1/2)     ]
                   [  - - 2          0      - 2          ]
                   [    6                   6            ]
                   [                                     ]
                   [1  (1/2)  (1/2)       1  (1/2)  (1/2)]
                   [- 2      3       0  - - 2      3     ]
                   [6                     6              ]

Bug Report Submitted

Lark — Sat, 20 Mar 2010 01:18:11 Z

I have added this report to our bugs database. Note that it is also possible to submit bug reports using the submit "Maple Software Change Request" link on the left-hand side of the page under Navigation.

No! The theorem you stated

brauersuzuki — Thu, 18 Mar 2010 22:09:31 Z

No! The theorem you stated is not an if and only if! It only says, IF the eigenvalues are distinct, THEN the matrix is diagonalizable. And not the other way round.

You may read

http://en.wikipedia.org/wiki/Symmetric_matrix

there you can find among other things:

"Every real symmetric matrix A can be diagonalized"

oops

PatrickT — Fri, 19 Mar 2010 01:16:25 Z

oops, sorry

multiplicity: algebraic vs geometric

Axel Vogt — Fri, 19 Mar 2010 01:22:00 Z

The actual structure of the JNF is given through algebraic (=roots of char polynomial counted with multiplicity) vs geometric multiplicity (=dimension of solutions for A - lambda*id ) for the Eigenvalues (though I do not have a reasonable reference at hand and after forgetting the precise statement after some time have to look it up in older notes again)