Understanding Idempotent Matrices and Matrix Equations

In linear algebra, an idempotent matrix is a square matrix that when multiplied by itself produces itself. That is, if A is an idempotent matrix, then A^2 A. Similarly, a matrix B is idempotent if B^2 B. This property is crucial in various fields such as computer science, statistics, and engineering.

Problem Statement

The problem at hand is to find idempotent matrices A and B such that AB A and BA B. This pair of conditions might seem restrictive, but they impose interesting properties on the matrices involved.

Analysis of Matrix Equations

Analysis of AB A

Starting with the equation AB A, we can rewrite it as:

AB - A 0

This can be further rewritten as:

AB - I 0

From this, it follows that the columns of A must lie in the null space of B - I. This means that A and the matrix B - I must be such that their multiplication results in a zero matrix.

Analysis of BA B

Similarly, from the equation BA B, we can rewrite it as:

BA - B 0

This can be further rewritten as:

B - IA 0

This implies that the columns of B must lie in the null space of A - I. Again, the matrices A and B must be such that their multiplication results in a zero matrix.

Conclusion for Matrices A and B

From the above analysis, we can conclude that:

The range of A must be contained within the null space of B - I. The range of B must be contained within the null space of A - I.

Some Possible Solutions

Identity Matrices

The most straightforward solution is when both A and B are identity matrices, i.e., A I and B I. In this case, both conditions are trivially satisfied because:

I cdot I I quad and quad I cdot I I

Zero Matrices

Another trivial solution is when both A and B are zero matrices, i.e., A 0 and B 0. This is because:

0 cdot 0 0 quad and quad 0 cdot 0 0

Non-Identity Matrices

It is possible to find non-identity idempotent matrices A and B that satisfy these conditions, but they would typically need to be structured in such a way that their ranges and null spaces align appropriately. For instance, projection matrices can be used, but the general solution would likely require specific constraints or specific forms for A and B.

Further Exploration

Further exploration of non-identity idempotent matrices could involve specific structural constraints. Projection matrices are a good starting point, but the combined condition of AB A and BA B typically leads back to the trivial cases without additional constraints.

Summary

In conclusion, the primary straightforward solutions to the problem are:

A I quad and quad B I A 0 quad and quad B 0

Other non-trivial solutions would likely require more constraints or specific forms for A and B. This property of idempotent matrices and their multiplication is an interesting and important concept in linear algebra.