Understanding Nilpotent Matrices with an Index of 1

Nilpotent matrices are a fascinating topic in linear algebra, and one specific aspect of interest is the index of nilpotency. The index of nilpotency of a nilpotent matrix is the smallest positive integer (n) such that (A^n 0). This article explores the special case where the index of nilpotency is 1 and why the only matrix with this property is the zero matrix.

Definition and Properties of Nilpotent Matrices

A square matrix (A) is considered nilpotent if there exists a positive integer (p) such that (A^p 0). When (p) is the smallest such integer, we say that (A) is nilpotent of index (p).

The key point to understand is that the index of nilpotency being 1 means that the matrix raised to the first power is the zero matrix. That is, for a nilpotent matrix (A) with index 1, (A 0).

The Zero Matrix

The zero matrix, denoted as (0), is the matrix where all entries are zero. For this matrix, raising it to any power will still yield the zero matrix because adding zeros together any number of times results in zero. Specifically, for the zero matrix (A), we have:

[A^1 A 0]

This makes it the only matrix that is nilpotent with an index of 1. If (A) is any (n times n) non-zero matrix, it cannot satisfy (A 0), and thus (A) will not be nilpotent with index 1.

Examples and Implications

Let's consider a simple example to solidify our understanding. Take the zero matrix (A begin{pmatrix} 0 0 0 0 end{pmatrix}). This matrix is clearly nilpotent with index 1 because:

[A^1 A begin{pmatrix} 0 0 0 0 end{pmatrix}]

Let's also consider a non-zero nilpotent matrix, such as:

[A begin{pmatrix} 0 1 0 0 end{pmatrix}]

This matrix is nilpotent but does not have index 1 because:

[A^2 begin{pmatrix} 0 1 0 0 end{pmatrix} begin{pmatrix} 0 1 0 0 end{pmatrix} begin{pmatrix} 0 0 0 0 end{pmatrix}]

In this case, the index of nilpotency is 2, not 1.

Conclusion

In conclusion, the zero matrix is the only matrix that is nilpotent with an index of 1. Any non-zero matrix will either not be nilpotent or will have a higher index of nilpotency. This property is important in various areas of mathematics, such as linear algebra, differential equations, and representation theory.

By understanding the concept of the index of nilpotency, particularly when it is 1, we gain insight into the unique properties of the zero matrix and its significance in the study of nilpotent matrices.

References

For further reading, you may want to explore the following resources:

Nilpotent Matrix on Wikipedia Nilpotent Matrix on MathWorld Nilpotent Matrices on Math Stack Exchange