Problem 17 - Advanced Engineering Mathematics

Question

A nonzero \(n \times n\) matrix \(A\) is said to be nilpotent of index \(m\) if \(m\) is the smallest positive integer for which \(\mathbf{A}^{m}=\mathbf{0}\). Which of the following matrices are nilpotent? If nilpotent, what is its index? (a) \(\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right)\) (b) \(\left(\begin{array}{rr}2 & 2 \\ -2 & -2\end{array}\right)\) (c) \(\left(\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 3 & 0\end{array}\right)\) (d) \(\left(\begin{array}{lll}0 & 0 & 5 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)\) (e) \(\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ -1 & 0 & 0 & 0\end{array}\right)\) (f) \(\left(\begin{array}{llll}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0\end{array}\right)\)

Step-by-Step Solution

Answer

Matrices (b), (c), (d), (e), and (f) are nilpotent with indices 2, 3, 2, 3, and 4, respectively.

Step 1: Understand the Given Matrix

Consider the matrix provided in part (a): \[ A = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} \]We need to check if there exists a positive integer \( m \) such that \( A^m = 0 \).

Step 2: Compute Powers of Matrix (a)

First, we compute \( A^2 = A \cdot A \):\[A^2 = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + 0 \cdot -1 & 1 \cdot 0 + 0 \cdot 0 \ -1 \cdot 1 + 0 \cdot -1 & -1 \cdot 0 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix}.\]Since \( A^2 = A eq 0 \), we check the next power, \( A^3 \):\[A^3 = A^2 \cdot A = A \cdot A = A^2 = \begin{pmatrix} 1 & 0 \ -1 & 0 \end{pmatrix} eq 0.\]Since \( A^3 eq 0 \), \( A \) is not nilpotent.

Step 3: Check Matrix (b) for Nilpotency

Consider the matrix provided in part (b): \[ B = \begin{pmatrix} 2 & 2 \ -2 & -2 \end{pmatrix} \]Let's calculate \( B^2 \):\[B^2 = \begin{pmatrix} 2 & 2 \ -2 & -2 \end{pmatrix} \begin{pmatrix} 2 & 2 \ -2 & -2 \end{pmatrix} = \begin{pmatrix} 2\cdot2 + 2\cdot-2 & 2\cdot2 + 2\cdot-2 \ -2\cdot2 + -2\cdot-2 & -2\cdot2 + -2\cdot-2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}.\]Since \( B^2 = 0 \), the matrix is nilpotent with an index of \( m = 2 \).

Step 4: Check Matrix (c) for Nilpotency

Consider the matrix in part (c): \[ C = \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} \]We need to compute powers to find when \( C^m = 0 \). Let's calculate \( C^2 \):\[C^2 = \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 3 & 0 & 0 \end{pmatrix} \]\[C^3 = C^2 \cdot C = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 3 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \ 1 & 0 & 0 \ 2 & 3 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}.\]Since \( C^3 = 0 \), the matrix is nilpotent with an index of \( m = 3 \).

Step 5: Check Matrix (d) for Nilpotency

Consider the matrix in part (d):\[ D = \begin{pmatrix} 0 & 0 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} \]Since this is a triangular matrix with the main diagonal elements as 0, if all products yield a zero matrix, it's nilpotent. Calculate \( D^2 \):\[D^2 = \begin{pmatrix} 0 & 0 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 5 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}.\]Since \( D^2 = 0 \), the matrix is nilpotent with index \( m = 2 \).

Step 6: Check Matrix (e) for Nilpotency

Consider the matrix provided in part (e):\[ E = \begin{pmatrix} 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ -1 & 0 & 0 & 0 \end{pmatrix} \]This is a 4x4 matrix; let's compute \( E^2 \):After computing, \[ E^2 = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \end{pmatrix} \]Compute \( E^3 \):\[ E^3 = E^2 \cdot E = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ -1 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 \ -1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix}.\]Since \( E^3 = 0 \), the matrix is nilpotent with index \( m = 3 \).

Step 7: Check Matrix (f) for Nilpotency

Consider the matrix in part (f):\[ F = \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} \]This is a 4x4 matrix. Compute \( F^2 \):After computing, \( F^2 = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 5 & 1 & 0 & 0 \end{pmatrix} \).Compute \( F^3 \):\[ F^3 = F^2 \cdot F = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 5 & 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{pmatrix} \].Compute \( F^4 \):\[ F^4 = F^3 \cdot F = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 \ 3 & 1 & 0 & 0 \ 2 & 2 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} \]Since \( F^4 = 0 \), the matrix is nilpotent with an index of \( m = 4 \).

Key Concepts

Nilpotency Index

The concept of a nilpotent matrix revolves around determining the smallest power, denoted as the nilpotency index, that results in the zero matrix. For a matrix to be nilpotent, there must exist a positive integer \( m \) such that \( A^m = 0 \), where \( A \) is the given matrix and \( 0 \) is the zero matrix of the same dimension. The value \( m \) is known as the nilpotency index, the smallest such power where this condition is met.
In the exercise, you compute different powers of given matrices to check whether they eventually result in a zero matrix. For instance, if \( B^2 = 0 \) for matrix \( B \), then \( B \) is nilpotent with an index of 2. Each matrix's nilpotency index provides insight into its behavior and properties within linear transformations.

Matrix Powers

Calculating matrix powers involves multiplying the matrix by itself repeatedly. This process continues until you reach a power resulting in a zero matrix when determining nilpotency. It's essentially examining how the transformation represented by the matrix evolves with repeated application.
The exercise illustrates this by starting from \( A^2 \), \( A^3 \), and so on, until a zero matrix appears using the smallest integer power possible. The moment you reach a zero matrix, the power used defines the matrix's nilpotency index. This is important because different matrices exhibit nilpotency at different levels, as seen from the examples where some become zero after squaring, others require cubing, and some need even more powers.

Triangular Matrix

Triangular matrices, whether upper or lower, are efficient when it comes to calculating powers due to their structure. An important property of some triangular matrices is that when they have zeros along the main diagonal and all positions above or below it are filled, they might turn nilpotent easily.
For example, consider matrix (d) from the exercise, which is a triangular matrix with all zero entries on the diagonal. In such matrices, the power required to reach a zero matrix is often lower, since zeros along the main diagonal imply the matrix naturally stabilizes to zero with fewer multiplications. This concept simplifies identifying nilpotency since such patterns often provide quick insights without extensive calculations.

Linear Algebra

Linear algebra provides the backbone for dealing with vector spaces and linear transformations. Here, we focus on matrices as representations of linear operators. Nilpotency is a concept within linear algebra that describes certain transformations that eventually lead to the zero transformation when applied repeatedly.
Understanding nilpotency is crucial as it helps in comprehending the solvability of algebraic systems, characterizing eigenvalues, and in simplifying complex matrix functions. In advanced linear algebra, knowing whether a transformation is nilpotent can lead to simplifications in computations, such as in the context of matrix exponentials and Jordan canonical forms. Through this exercise, students get a foundational grasp of these properties, enhancing their understanding of matrix behaviors in transformations.