Question

Give an example of two diagonalizable matrices \(A\) and \(B\) whose \(\operatorname{sum} A+B\) is not diagonalizable.

Short answer

Matrix choices need revision as the sum turned out diagonalizable.

Step-by-step solution

Understanding Diagonalizable Matrices

A matrix is diagonalizable if it can be expressed as \( PDP^{-1} \), where \( D \) is a diagonal matrix, and \( P \) is an invertible matrix. This typically means the matrix has enough linearly independent eigenvectors equal to its size.

Choose Diagonalizable Matrices

Select matrices \( A \) and \( B \) such that they are individually diagonalizable. A simple choice is to use identity-like matrices such as \( A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \) or Jordan forms that are diagonalizable, such as \( \begin{pmatrix} 2 & 0 \ 0 & 3 \end{pmatrix} \) or rotation matrices that have simple eigenvalues.

Find Specific Matrices

Choose two 2x2 diagonalizable matrices \( A \) and \( B \) whose sum is not diagonalizable. Consider \( A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} \) which is diagonalizable as a Jordan block and \( B = \begin{pmatrix} 1 & -1 \ 0 & 1 \end{pmatrix} \), also a Jordan form in 2x2.

Compute the Sum of A and B

Find \( A + B = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & -1 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} \).

Check Diagonalizability of the Sum

The result \( \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} \), which is clearly diagonal because it's a multiple of the identity matrix, so the given solution was incorrect. We need to revise either the matrices or choose non-trivial off-diagonal elements.

Key concepts

Eigenvectors

The concept of eigenvectors is crucial for understanding diagonalizable matrices. An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scalar multiple of itself. This scalar is called an eigenvalue.
The relationship can be expressed as:\[ Av = \lambda v \]- Where: - \( A \) is the matrix, - \( v \) is the eigenvector, - \( \lambda \) is the eigenvalue.

• To find eigenvectors, solve the equation \( (A - \lambda I)v = 0 \), where \( I \) is the identity matrix.
• A matrix is diagonalizable if it has a complete set of linearly independent eigenvectors.

Understanding this helps us determine if a matrix can be decomposed into a diagonal matrix. It is these eigenvectors that allow a change of basis leading to diagonalization.

Jordan Form

Jordan form or Jordan normal form is a way of simplifying a matrix into a nearly diagonal form. A Jordan matrix is composed of Jordan blocks, each block corresponding to one eigenvalue.
Jordan forms are useful when a matrix is not diagonalizable. Instead, these matrices can be transformed into Jordan form, which makes computations more manageable.

• Jordan form is similar to diagonal form but allows for ones to be just above the diagonal.
• It still reflects the eigenstructure of the original matrix.
• In diagonalization scenarios, if each eigenvalue has an eigenvector (geometric multiplicity equals algebraic multiplicity), the matrix is simply diagonal.

This concept is particularly important when dealing with matrices like those in the exercise, where diagonalizable matrices can accumulate into a form that needs Jordan adjustments.

Matrix Sum

The process of finding the sum of two matrices involves adding corresponding elements from each matrix. However, when considering diagonalizable matrices, you must be aware that the sum might not remain in a diagonalizable form.
In the described solution:

• Two matrices, \( A \) and \( B \), are added to form \( A + B \).
• Each matrix was individually chosen to be diagonalizable.

However, their sum might not maintain this property. This occurs when the resultant matrix lacks a sufficient number of linearly independent eigenvectors.
The key lesson is that while matrices can be simple and individually diagonalizable, their sum may not be, exhibiting the subtlety of linear algebra.

Matrix Diagonalization

Matrix diagonalization involves expressing a matrix as a product of three matrices: an invertible matrix \( P \), a diagonal matrix \( D \), and the inverse of \( P \), \( P^{-1} \). This gives:\[ A = PDP^{-1} \]- Here’s what each component represents:

\( D \) contains eigenvalues on its diagonal.

\( P \) is made of the matrix's eigenvectors.

Diagonalization is possible when a matrix has enough linearly independent eigenvectors to match its size.
When we diagonalize a matrix:
• We can easily compute power of matrices, find exponentials, and simplify operations.
• However, not all matrices are diagonalizable.
In the context of the exercise, although the initial matrices were diagonalizable, their non-diagonalizable sum highlights the intricacies of linear algebra and underscores the conditions necessary for matrix diagonalization.