Prove that A and B are simultaneously diagonalizable.

  • Thread starter Daron
  • Start date
In summary, A and B are commuting diagonalizable matrices and it is proven that they are simultaneously diagonalizable by showing that they have a unique Jordan form and that there exists a matrix C such that CBC-1 = J(C) where J(C) is C's Jordan form. As C is unique, it is shown that A and B are simultaneously diagonalizable.
  • #1
Daron
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Homework Statement



A and B are commuting diagonalizable matrices. Prove that they are simultaneously diagonalizable.

Homework Equations



AB = BA

The Attempt at a Solution



I have what looks like a proof, but I'm not very happy with it. Is there anything wrong here?

AB = BA
B = ABA-1

Every matrix has exactly one jordan form. All diagonal matrices are jordan forms. So there exists a unique marix C such that
CBC-1 = J(C) where J(C) is C's Jordan form.
J(C) = CBC-1 = CABA-1C-1 = (CA)B(CA)-1
As C is unique, C = CA, so A = I
And CIC-1 = I, which is diagonal.
So A and B are simultaneously diagonalisable.
 
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  • #2
Can you assume A has an inverse?
 
  • #3
I figured it out.
 

FAQ: Prove that A and B are simultaneously diagonalizable.

What does it mean for two matrices to be simultaneously diagonalizable?

Two matrices, A and B, are said to be simultaneously diagonalizable if there exists a single invertible matrix P such that both P-1AP and P-1BP are diagonal matrices.

How can I prove that two matrices are simultaneously diagonalizable?

In order to prove that two matrices are simultaneously diagonalizable, you can use the following steps:

  1. Find the eigenvalues and corresponding eigenvectors for both matrices.
  2. Check if the eigenvectors for each matrix are linearly independent.
  3. If the eigenvectors are linearly independent, construct a matrix P with the eigenvectors as its columns.
  4. Calculate P-1AP and P-1BP, and check if both are diagonal matrices.
  5. If both P-1AP and P-1BP are diagonal matrices, then A and B are simultaneously diagonalizable.

Can two non-commuting matrices be simultaneously diagonalizable?

No, two non-commuting matrices cannot be simultaneously diagonalizable. This is because if two matrices do not commute, it is not possible to find a single matrix P that simultaneously diagonalizes both of them.

What are the benefits of having two matrices be simultaneously diagonalizable?

Having two matrices be simultaneously diagonalizable allows for easier calculations and simplification of expressions involving the matrices. It also provides a convenient way to find the powers of a matrix and to solve systems of linear equations.

Is simultaneous diagonalization possible for more than two matrices?

Yes, simultaneous diagonalization is possible for more than two matrices. In general, a set of n matrices can be simultaneously diagonalizable if and only if they commute with each other.

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