Diagonal, Jordan Normal and And Inverses

1. May 24, 2009

heshbon

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. May 24, 2009

heshbon

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

3. May 24, 2009

heshbon

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. May 24, 2009

Random Variable

1) No. Let $$d_{i}$$ be the diagonal entries of a diagonal matrix The inverse (if it exists) consists of the values $$\frac {1}{d_{i}}$$ along the diagonal. Therefore, all $$d_{i}$$'s must be nonzero for the inverse to exist.

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

3) I don't know. :uhh:

5. May 24, 2009

HallsofIvy

Staff Emeritus
Re: Jordan Normal, Diagonal and Inverses

Is
$$\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 0\end{bmatrix}$$
invertible?

6. May 26, 2009

heshbon

Thanks greatly.

That clears it all up.