Problem 20 - Differential Equations and Linear Algebra

Question

Let \(A\) and \(B\) be \(n \times n\) matrices. Let \(v\) be an eigenvector of \(A\) corresponding to \(\lambda_{1}\) and let \(v\) also be an eigenvector of \(B\) corresponding to eigenvalue \(\lambda_{2} .\) Show that \(A B-B A\) is not invertible.

Step-by-Step Solution

Answer

Given that v is an eigenvector of both A and B, we have the relationships \(A v = \lambda_1 v\) and \(B v = \lambda_2 v\). Rewriting the expression \((A B - B A) v = \lambda_1 v - \lambda_1 \lambda_2 v\), we can show that the null space of AB - BA is non-empty, as v is a non-zero vector that lies in the null space. Therefore, AB - BA is singular, which means it is not invertible.

Step 1: Recall eigenvector and eigenvalue properties

An eigenvector v of matrix A satisfies \(A v = \lambda_1 v\) where λ1 is the eigenvalue corresponding to the eigenvector v. Similarly, for matrix B, it satisfies \(B v = \lambda_2 v\).

Step 2: Observe properties of the matrices A and B

Given that v is an eigenvector of both A and B, we have the following relationships: \(A v = \lambda_1 v\) \(B v = \lambda_2 v\) Additionally, since λ1 and λ2 are scalars, we can multiply the second equation by λ1 and subtract it from the first one, which gives:

Step 3: Subtract scaled Eigenvalue equations

Subtract the second equation multiplied by λ1 from the first equation: \(A v - \lambda_1 B v = \lambda_1 v - \lambda_1 \lambda_2 v\).

Step 4: Rewrite the equation using commutator

Factor v from the above equation and rewrite the left-hand side as a commutator: \((A B - B A) v = \lambda_1 v - \lambda_1 \lambda_2 v\)

Step 5: Show that the commutator has a non-empty null space

We can rewrite the above equation as: \((A B - B A) v = (\lambda_1 - \lambda_1 \lambda_2) v\) Since v is non-zero (by definition of eigenvector), it means that v lies in the null space of AB - BA. Hence, the null space of AB - BA is non-empty.

Step 6: Conclude that the commutator is not invertible

Since the null space of AB - BA is non-empty, it means that AB - BA is singular, indicating it is not invertible.

Key Concepts

Eigenvectors and Eigenvalues

In linear algebra, the concepts of _eigenvectors_ and _eigenvalues_ are fundamental for understanding how matrices operate. An eigenvector of a square matrix is a non-zero vector that changes at most by a scalar factor when that matrix is applied to it. That scalar is called an eigenvalue. For a matrix \(A\), if \(v\) is an eigenvector and \(\lambda\) is its corresponding eigenvalue, they satisfy the relationship:

• \(A v = \lambda v\)

This is a cornerstone concept used in many areas, such as systems of linear equations, stability analysis, and quantum mechanics.
Eigenvectors and eigenvalues allow us to decompose a matrix into simpler parts, making complex operations more manageable. They provide insight into the properties of the matrix and can reveal underlying structures in linear transformations. It is important to understand how different matrices can share eigenvectors, which, as in our exercise, implies strong interactions between the operations they represent.

Matrix Commutator

The _matrix commutator_ gives us a measure of how one matrix transformation is affected by another. For two matrices \(A\) and \(B\), the commutator \([A, B]\) is defined as:

• \([A, B] = AB - BA\)

If \([A, B] = 0\), the matrices \(A\) and \(B\) are said to commute; otherwise, they do not.
Understanding matrix commutators is crucial in physics, particularly in quantum mechanics, where non-commuting operators reflect fundamental uncertainty principles.
In our exercise, \(v\) being an eigenvector of both \(A\) and \(B\) helps in understanding the properties of the commutator \(AB - BA\). When we analyzed it with respect to \(v\), it demonstrated that the null space of the commutator is non-empty, exercising its impact on invertibility.

Invertibility of Matrices

A matrix is said to be _invertible_ if there exists another matrix such that their product is the identity matrix. Inversion is an essential operation in solving linear systems, mathematical modeling, and inverse problems. An invertible matrix must be square and non-singular, meaning it should have full rank with no empty null space.
In our exercise, we conclude that the commutator \(AB - BA\) is not invertible. When the null space of a matrix is non-empty, this means there exist non-zero vectors that are mapped to zero by the matrix, indicating the matrix is singular. The matrix \(AB - BA\) has such a null space since the eigenvector \(v\) satisfies \((AB - BA) v = 0\) with \(veq0\).
Therefore, understanding this property of inversion connects eigenvalues and eigenvectors to deeper insights into the nature of matrices, providing practical insights into when operations between matrices become complex.