Results 1 to 2 of 2

Math Help - Eigenvalues

  1. #1
    Member
    Joined
    Feb 2009
    From
    Chennai
    Posts
    148

    Eigenvalues

    Let A be an n \times n real matrix such that A^{2}=I, A \neq \pm{I}
    (where I denotes the n n-identity matrix). Show that
    (i) A has two eigenvalues  \lambda_{1},\lambda_{2}.
    (ii) Every element x \in \mathbb{R}^{n} can be expressed uniquely as x_{1}+x_{2},
    where Ax_{1}=\lambda_{1}x_{1} and Ax_{2}=\lambda_{2}x_{2}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Mar 2009
    From
    Grenoble
    Posts
    5
    Quote Originally Posted by Chandru1 View Post
    Let A be an n \times n real matrix such that A^{2}=I, A \neq \pm{I}
    (where I denotes the n n-identity matrix). Show that
    (i) A has two eigenvalues  \lambda_{1},\lambda_{2}.
    (ii) Every element x \in \mathbb{R}^{n} can be expressed uniquely as x_{1}+x_{2},
    where Ax_{1}=\lambda_{1}x_{1} and Ax_{2}=\lambda_{2}x_{2}
    Take two endomorphisms of \mathbb{R}^{n} :
    u = A - I
    v = A + I

    You have

    • u \circ v = v \circ u = 0
    • u \neq 0
    • v \neq 0

    according to the problem.

    You see that

    • Ker (u) \cap Ker (v) = \oslash
    • \mathbb{R}^{n} = Ker (u \circ v) = Ker (v) \cup v^{-1}(Ker (u) \cap Im (v))

    and, after a little manipulation and comaprison of dimensions,
    dim Ker (u) + dim Ker (v) = n

    Which is sufficient to say that Ker(u) \oplus Ker(v) = \mathbb{R}^{n}

    dim Ker (u) \neq 0, cause it would mean v = 0
    dim Ker (v) \neq 0, cause it would mean u = 0

    So there are effectively 2 eigenvalues for A ( 1 and -1),
    and the associated eigenspaces are in direct sum (Ker u and Ker v),
    which guarantees the existance and unicity if the decomposition.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 21st 2010, 01:57 PM
  2. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 21st 2009, 01:38 PM
  3. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 13th 2008, 08:43 AM
  4. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 12th 2008, 11:12 PM
  5. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 12th 2008, 08:25 PM

Search Tags


/mathhelpforum @mathhelpforum