Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By Deveno

Math Help - Question about definition of the term "diagonally dominant eigenvalue"

  1. #1
    Member
    Joined
    Apr 2011
    From
    Somwhere in cyberspace.
    Posts
    101
    Thanks
    4

    Question about definition of the term "dominant eigenvalue"

    I need help. Consider a matrix that has only one eigenvalue (say 3) but this value has algebraic multiplicity bigger than 1.
    Is 3 the dominant eigenvalue of the matrix or not? When determining if there is dominant eigenvalue,
    how do we "count" eigenvalues? As 3, 3 or just one 3?

    Edit: Such a matrix can be for example:

    | 3 1 |
    | 0 3 |

    Edit: The topic of the thread - "diagonally dominant eigenvalue" is a mistake and cannot be edited. Please disregard "diagonally"
    Last edited by mrproper; January 30th 2013 at 08:12 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    Re: Question about definition of the term "diagonally dominant eigenvalue"

    um,... yes. an eigenvalue is just an eigenvalue. the multiplicity of an eigenvalue only comes into it when you are considering which eigenvectors belong to that eigenvalue. eigenvalues with algebraic multiplicity may split into multiple linearly independent eigenvectors (or may not). if a matrix has but a single eigenvalue, yes, that eigenvalue is the dominant one, by default.

    one caveat, however: a real nxn matrix may have complex eigenvalues which dominate the real ones. for practical applications, it is often assumed the matrix is diagonalizable over the field being used.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Apr 2011
    From
    Somwhere in cyberspace.
    Posts
    101
    Thanks
    4

    Re: Question about definition of the term "dominant eigenvalue"

    So, the matrix

    | 3 0 |
    | 0 3 |

    has dominant eigenvalue. That is 3.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    Re: Question about definition of the term "diagonally dominant eigenvalue"

    yes.
    Thanks from mrproper
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: April 2nd 2012, 08:06 PM
  2. Replies: 3
    Last Post: September 23rd 2011, 06:43 AM
  3. Replies: 1
    Last Post: October 25th 2010, 05:45 AM
  4. Rigorous Definition of "Inequality" or "Positive" in the Reals
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: July 22nd 2010, 01:23 AM
  5. A question about definition of "Binary Relation"
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: October 16th 2009, 03:56 AM

Search Tags


/mathhelpforum @mathhelpforum