# Diagonalizable Proof Homework: True or False?

• metder
In summary, the conversation discusses the linear operators T and U on the space of n X n matrices over F. The question of whether or not T and U are diagonalizable when A is diagonalizable is posed. The answer is that T is diagonalizable, while U is not necessarily diagonalizable. The conversation also mentions that A being diagonalizable is not the same as T and U being diagonalizable, as they are linear operators rather than simple matrices.
metder

## Homework Statement

Let V be the space of n X n matrices over F. Let A be a fixed n X n matrix
over F. Let T and U be the linear operators on V defined by
T(B) = AB
U(B) = AB - BA.
1. True or false? If A is diagonalizable (over F), then T is diagonalizable.
2. True or false? If A is diagonalizable, then U is diagonalizable
Thanks for the help.

## The Attempt at a Solution

I'm guessing that 1 is true and 2 is false. I'm not sure, since these are linear operators rather than simple matrices.

Consider, first, the case in which A is diagonal. What do T and U do to the "basis" matrices?

## 1. What is the definition of diagonalizable?

Diagonalizable refers to a mathematical property of a square matrix, where it can be transformed into a diagonal matrix through a similarity transformation. This means that the matrix has distinct eigenvalues and a set of linearly independent eigenvectors.

## 2. How do you prove if a matrix is diagonalizable or not?

To prove if a matrix is diagonalizable, you need to check if it has a full set of linearly independent eigenvectors. This can be done by finding the eigenvalues and corresponding eigenvectors of the matrix. If the number of distinct eigenvalues is equal to the dimension of the matrix, then it is diagonalizable.

## 3. Is every square matrix diagonalizable?

No, not every square matrix is diagonalizable. A matrix is only diagonalizable if it has a full set of linearly independent eigenvectors. If the matrix has repeated eigenvalues or does not have enough eigenvectors, then it is not diagonalizable.

## 4. Can a non-square matrix be diagonalizable?

No, a non-square matrix cannot be diagonalizable. The diagonalizability property only applies to square matrices.

## 5. Can a matrix be diagonalizable over any field?

No, not all matrices are diagonalizable over any field. A matrix is diagonalizable over a field if and only if it has a full set of linearly independent eigenvectors in that field. This means that the field must contain the eigenvalues of the matrix.

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