Let A be a diagonalizable n x n matrix. Prove that if B is any matrix that is similar to A, then B is diagonalizable
If A is diagonalizable, then there exists a matrix P such that
D = P-1 A P
where D is a diagonal matrix.
If A and B are similar, then there exists a matrix Q such that:
A = Q-1 B Q
We can obtain diagonal matrix D by a second transformation such that:
D = Q-1 P-1 B P Q
QED
To prove that if B is any matrix similar to A, then B is diagonalizable, we need to show that B can be written in the form B = PDP^(-1), where D is a diagonal matrix and P is an invertible matrix.
Let's break down the steps to prove this:
1. Given that A is a diagonalizable n x n matrix, we know that there exists an invertible matrix P such that A = PDP^(-1), where D is a diagonal matrix.
2. Now, since B is similar to A, there exists an invertible matrix Q such that B = QAQ^(-1).
3. We want to show that B can be written in the form B = PDP^(-1), where D is a diagonal matrix and P is an invertible matrix. To do this, we can substitute A with its equivalent expression in terms of P and D: B = Q(PDP^(-1))Q^(-1).
4. Rearranging the expression, we have: B = (QP)D(P^(-1)Q^(-1)).
5. Since P and Q are invertible matrices, their product QP is also invertible.
6. Letting R = QP, we have B = RDR^(-1), where R is an invertible matrix.
7. Now, we can rewrite B in the desired form: B = PDP^(-1).
8. Therefore, we have shown that if B is any matrix that is similar to a diagonalizable matrix A, then B is also diagonalizable.
By following these steps, we have proved that if B is any matrix similar to A, then B is diagonalizable.