Product of Diagonalizable matrices

In summary, the conversation discusses the statement (2) which states that products of diagonalizable matrices are always diagonalizable. The individual studying for an exam has found a counterexample to this statement by using the square of the identity matrix. They also mention trying to construct a counterexample for 2x2 matrices, but have not been successful. They ask for hints and mention the definition of being diagonalizable as a possible approach.
  • #1
Klungo
136
1
Ignore post, I found a counterexample to (2).

I'm studying for an upcoming exam, and I'm a bit confused about how to go about proving or disproving the statement (2).

1.) Products of diagonalizable matrices are never diagonalizable.

I figured false and my counterexample is really just the square of the identity matrix. Since I is diagonalizable and I^2=I.

2.) Productions of diagonalizable matrices are always diagonalizable.

I'm not too sure if it's true to begin with. Trying to construct a counterexample for 2x2 matrices hasn't been successful yet. Any hints much appreciated.
 
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  • #2
A matrix, A, on Rn, is "diagonalizable" if and only if there exist a basis for Rn consisting of eigenvectors of A. Can you use that?
 

What is a product of diagonalizable matrices?

A product of diagonalizable matrices is the result of multiplying two or more matrices that are diagonalizable. It is also known as a diagonalizable matrix product.

What does it mean for a matrix to be diagonalizable?

A matrix is diagonalizable if it can be expressed as a product of two matrices, where one of the matrices is diagonal and the other is invertible. This means that the matrix can be simplified into a diagonal form, making it easier to work with.

Why is the product of diagonalizable matrices important?

The product of diagonalizable matrices is important because it allows us to simplify complex matrices and make calculations easier. It also has many applications in fields such as physics, engineering, and economics.

How do you know if two matrices are diagonalizable?

Two matrices are diagonalizable if they have the same eigenvectors. This means that they can be simplified into diagonal form and multiplied together to get the product of diagonalizable matrices.

What are some properties of the product of diagonalizable matrices?

Some properties of the product of diagonalizable matrices include the fact that the product of two diagonalizable matrices is also diagonalizable, and that the eigenvalues of the product are the products of the eigenvalues of the individual matrices. Additionally, the product of diagonalizable matrices is commutative, meaning that the order in which the matrices are multiplied does not affect the result.

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