Question

Show that every eigenvalue of \(A\) is zero if and only if \(A\) is nilpotent \(\left(A^{k}=0\right.\) for some \(\left.k \geq 1\right)\).

Short answer

Every eigenvalue of \(A\) is zero if and only if \(A\) is nilpotent, as shown using properties of eigenvalues and characteristic polynomials.

Step-by-step solution

Clarify Nilpotent Matrix

A matrix \(A\) is nilpotent if there exists an integer \(k\geq 1\) such that \(A^k = 0\). This means that multiplying the matrix by itself some number of times results in a zero matrix.

Understand Eigenvalues of a Nilpotent Matrix

If \(A\) is nilpotent, then by definition \(A^k = 0\) for some positive integer \(k\). Hence, the characteristic polynomial of \(A\) is \(\lambda^n\), since \(A^k = P(A) = 0\) implies all roots (eigenvalues) of \(A\) are zero due to \(\lambda^k = 0\). Thus, all eigenvalues of a nilpotent matrix must be zero.

Suppose All Eigenvalues are Zero

Assume all eigenvalues of \(A\) are zero, then the characteristic polynomial of \(A\) is \(\lambda^n\). This implies the minimal polynomial \(m(\lambda)\) divides \(\lambda^n\), and is in the form \(\lambda^m\) where \(m \leq n\). Thus, \(A^m = 0\) and \(A\) is nilpotent.

Establish 'If and Only If' Connection

We have shown two things: If \(A\) is nilpotent, then all eigenvalues are zero (Step 2). Conversely, if all eigenvalues are zero, \(A\) is nilpotent (Step 3). Therefore, every eigenvalue of \(A\) is zero if and only if \(A\) is nilpotent.

Key concepts

Eigenvalues

Eigenvalues of a matrix are special numbers that give us important information about the matrix's properties. When you calculate the eigenvalues of a matrix \(A\), you are essentially solving for \(\lambda\) in the equation \(A\mathbf{v} = \lambda\mathbf{v}\), where \(\mathbf{v}\) is an eigenvector. Here are some key points to remember about eigenvalues:

• For a nilpotent matrix \(A\), all eigenvalues are zero. This is evident because if \(A^{k} = 0\) for some \(k\), the only value of \(\lambda\) that satisfies the characteristic equation is zero.
• The eigenvalues give us an idea of the matrix’s behavior over successive powers. For nilpotent matrices, since all eigenvalues are zero, the matrix repeatedly applied to itself results in the zero matrix.
• Understanding the relationship between eigenvalues and matrix types, like nilpotent matrices, helps in analyzing stability and system behavior in applied mathematics.

It’s crucial to grasp the concept that eigenvalues being zero ties directly to a matrix being nilpotent, as they are interconnected.

Characteristic Polynomial

The characteristic polynomial of a matrix is a polynomial equation that is fundamental in finding the eigenvalues of a matrix. For an \(n \times n\) matrix \(A\), the characteristic polynomial is defined as \(\det(\lambda I - A)\), where \(I\) is the identity matrix of the same size as \(A\) and \(\lambda\) is a scalar. Here’s why it matters:

• The roots of the characteristic polynomial are the eigenvalues of the matrix. For a nilpotent matrix, the characteristic polynomial looks like \(\lambda^n\), with all terms equating to zero, thereby making zero the only eigenvalue.
• The degree of the polynomial matches the size of the matrix, and knowing the characteristic polynomial can help determine the matrix's other properties, such as rank and whether it's invertible.
• The characteristic polynomial reveals important clues about the nature of the transformations a matrix represents, particularly in systems where the dynamics are dictated by matrix powers.

Understanding the characteristic polynomial is integral when analyzing nilpotent matrices because it directly determines the nature of the eigenvalues.

Minimal Polynomial

The minimal polynomial of a matrix is the smallest degree monic polynomial which the matrix satisfies, providing insights into the matrix's structure. For a matrix \(A\), the minimal polynomial will divide any other polynomial the matrix satisfies, including the characteristic polynomial. Some insights are as follows:

• For a nilpotent matrix, the minimal polynomial takes the form \(\lambda^m\), where \(m\) is the smallest integer such that \(A^m = 0\). This equation tells us that multiplying \(A\) by itself \(m\) times results in a zero matrix.
• The minimal polynomial can also provide information regarding the matrix’s block structure in its Jordan form, indicating possible simplifications in its representation.
• Due to its divisibility property, understanding the minimal polynomial helps not only in establishing matrix powers but also in deciding decompositions used in more advanced matrix theory applications.

Thus, the minimal polynomial is invaluable when demonstrating that a matrix is nilpotent, clarifying the conditions under which the matrix will become null.