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Diagonal, Jordan Normal and And Inverses

  1. May 24, 2009 #1
    1) Is a diagonal matrix always invertable?
    2) Is an Invertable matrix always Diagonalizable?
    3) Is a matrix in jordan normal form always invertable

    The answers are prob straight foward but im confused.
     
  2. jcsd
  3. May 24, 2009 #2
    Jordan Normal, Diagonal and Inverses

    1. The problem statement, all variables and given/known data

    1) Is a diagonal matrix always invertable?
    2) Is an Invertable matrix always Diagonalizable?
    3) Is a matrix in jordan normal form always invertable

    The answers are prob straight foward but im confused.

    2. Relevant equations



    3. The attempt at a solution
     
  4. May 24, 2009 #3
    I've had a thought about this, I guess what im looking for is some sort of relation between the characteristic polinomial and the determinant of a the matrix.
     
  5. May 24, 2009 #4
    1) No. Let [tex] d_{i} [/tex] be the diagonal entries of a diagonal matrix The inverse (if it exists) consists of the values [tex] \frac {1}{d_{i}} [/tex] along the diagonal. Therefore, all [tex] d_{i}[/tex]'s must be nonzero for the inverse to exist.

    2) No. If an nxn matrix is diagonalizable, it must have n linearly independent eigenvectors. [tex] A = \left(\begin{array}{cc}1&0\\1&1\end{array}\right) [/tex] is invertibile but not diagonalizable.

    3) I don't know. :uhh:
     
  6. May 24, 2009 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: Jordan Normal, Diagonal and Inverses

    Is
    [tex]\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 0\end{bmatrix}[/tex]
    invertible?
     
  7. May 26, 2009 #6
    Thanks greatly.

    That clears it all up.
     
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