True or False Justify your answer If two square matrices A a

True or False? Justify your answer.

If two square matrices A and B represent the same linear transformation

T: V -> V
in two different bases in V, and A is not diagonalizable then also B is not diagonalizable.

Solution

We know that a linear operator T : V V on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis for V such that the matrix of T with respect to the basis is a diagonal matrix. Also, a square matrix A is called diagonalizable if the linear transformation representing A is diagonalizable. Thus, if the matrix A representing the linear transformation T: V V with respect to an ordered basis for V is not diagonalizable, then T is not diagonalizable so that is also not diagonalizable. The statement is true.