Prove that if A is diagonalizable then AT is also diagonaliz

Prove that if A is diagonalizable then A^T is also diagonalizable.

Solution

If A is diagonalizable, then it can be written in the form:

A = PDP* where D is a diagonal matrix and P is an invertible matrix (I\'m using * to denote the inverse)

Now take the transpose of both sides:

A^t = (PDP*)^t = (P*)^tD^tP^t since the transpose of a product is the product of the transposes in reverse order.
Since the transpose of an inverse equals the inverse of the transpose, we get that this equals:

A^t = (P^t)*D^tP^t

But the transpose of a diagonal matrix is also diagonal, so D^t is diagonal (and in fact equals D).
Thus we have that A transpose is diagonalizable.

(You may feel more comfortable if you rename the matrix (P^t)* as Q. Then Q* equals P^t. And D^t = D as noted above. So we can rewite the last line as:

A^t = QDQ* so it now has the same appearance as the original relation between A and its diagonal form)