hi,

what does it mean to say that a matrix A is diagonalizable iff R^n has a basis consisting of eigenvectors for A?

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- Nov 24th 2010, 07:47 AMalexandrabel90diagonalizable
hi,

what does it mean to say that a matrix A is diagonalizable iff R^n has a basis consisting of eigenvectors for A? - Nov 24th 2010, 09:14 AMFernandoRevilla
If is such a basis then,

If we denote

then

This means i.e. is similar to a diagonal matrix (definition of diagonalizable matrix).

Try to prove the reciprocal statement, if a is diagonalizable then ...

Regards.

Fernando Revilla - Nov 24th 2010, 09:17 AMalexandrabel90
so in this case, instead of B consisting of eigenvectors, B consists of elementary matrices?

- Nov 24th 2010, 10:15 AMFernandoRevilla
If is similar to a diagonal matrix , (that is there exists invertible such that ) try to prove that the columns of determine a basis of eigenvectors)

: is equivalent to .__Hint__

Regards.

Fernando Revilla - Nov 25th 2010, 03:32 AMHallsofIvy