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Diagonal, Jordan Normal and And Inverses

  • Thread starter heshbon
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  • #1
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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.
 

Answers and Replies

  • #2
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Jordan Normal, Diagonal and Inverses

Homework Statement



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.

Homework Equations





The Attempt at a Solution

 
  • #3
27
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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.
 
  • #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:
 
  • #5
HallsofIvy
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Is
[tex]\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 0\end{bmatrix}[/tex]
invertible?
 
  • #6
27
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Thanks greatly.

That clears it all up.
 

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