Question

An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{m}=0\) for some \(m \geq 1\) a. Show that every triangular matrix with zeros on the main diagonal is nilpotent. b. If \(A\) is nilpotent, show that \(\lambda=0\) is the only eigenvalue (even complex) of \(A\). c. Deduce that \(c_{A}(x)=x^{n},\) if \(A\) is \(n \times n\) and nilpotent.

Short answer

A triangular matrix with zeros on the diagonal is nilpotent. If nilpotent, its only eigenvalue is zero, and its characteristic polynomial is \(x^n\).

Step-by-step solution

Understanding Nilpotent Matrices

A nilpotent matrix is a square matrix for which some positive power of the matrix equals 0. In this exercise, \(m\) is such that \(A^m = 0\). We will use properties of matrices and eigenvalues to solve the given problems.

Show Triangular Matrix is Nilpotent

Consider an upper triangular matrix \(A\) with zeros on the main diagonal:\[A = \begin{pmatrix}0 & a_{12} & a_{13} & \cdots & a_{1n} \0 & 0 & a_{23} & \cdots & a_{2n} \\vdots & \vdots & \vdots & \ddots & \vdots \0 & 0 & 0 & \cdots & 0\end{pmatrix}\]Each diagonal entry being zero ensures that when you multiply the matrix by itself, each entry below the main diagonal will multiply the subsequent rows by zero repeatedly. Therefore, \(A^n = 0\), proving it is nilpotent.

Show Eigenvalue of Nilpotent Matrix

If a matrix \(A\) is nilpotent, then by definition \(A^m = 0\) for some \(m\). Therefore, for any eigenvalue \(\lambda\) with eigenvector \(v\), we have \(A^m v = \lambda^m v = 0\). Since the eigenvector is not zero, \(\lambda^m = 0\) implies \(\lambda = 0\), making zero the only eigenvalue.

Deduce Characteristic Polynomial

The characteristic polynomial of a square matrix \(A\) is given by \(c_A(x) = \det(xI - A)\). For an \(n \times n\) nilpotent matrix, since \(\lambda = 0\) is the only eigenvalue and it accounts for all \(n\) eigenvalues, \(c_A(x) = x^n\). This means that all terms of lower degree are zero, reinforcing that \(A^m = 0\) for some \(m\).

Key concepts

Triangular Matrix

Triangular matrices are a special type of square matrices. They can either be upper triangular or lower triangular. An upper triangular matrix has all its entries below the main diagonal (from the top left to the bottom right) equal to zero.
In contrast, a lower triangular matrix has all its entries above the main diagonal equal to zero.

In our exercise, we focus on an upper triangular matrix where all diagonal elements are zero. This means if you were to look along the main diagonal of the matrix, you would see only zeros.

This configuration is significant because multiplying such a matrix by itself eventually results in a matrix where all entries are zero. This happens because each diagonal element stays zero, and the non-diagonal elements effectively multiply through zero when the matrix is raised to its power, leading to a completely zero matrix at a certain power level.

• The entries along the diagonal being zero is crucial because it impacts how quickly the matrix becomes a zero matrix when raised to a power.
• The larger the matrix, the higher power you might need to raise it to achieve a zero matrix.

Eigenvalue of a Matrix

An eigenvalue is a special number associated with a matrix. Given a square matrix \(A\), if there exists a vector \(v\) (called an eigenvector) such that \( Av = \lambda v \), then \( \lambda \) is called an eigenvalue of \(A\).
Eigenvalues provide insight into the properties of a matrix.

When a matrix is nilpotent, which means some power of it equals zero, its eigenvalues must also have some particular properties. Specifically, if \(A^m = 0\), then all eigenvalues are zero. This is because for any eigenvalue \(\lambda\), when you apply \(A\) repeatedly as shown in the eigenvalue equation \(A^m v = \lambda^m v = 0\), it implies that \(\lambda^m = 0\).
This results in \(\lambda = 0\) being the only solution because an eigenvector \(v\) is non-zero, and dividing by \(v\) gives the result.

• Eigenvalues help to describe how the matrix behaves under linear mappings.
• For nilpotent matrices, \(\lambda = 0\) is always the eigenvalue.

Characteristic Polynomial

The characteristic polynomial of a matrix provides a way to encapsulate all the eigenvalues of a matrix in a single expression.

For an \(n \times n\) matrix \( A \), the characteristic polynomial \( c_A(x) \) is defined by the determinant \( \det(xI - A) \), where \( I \) is the identity matrix of the same size as \( A \).

This polynomial makes it possible to find all the eigenvalues by solving \( c_A(x) = 0 \).

• When a matrix is nilpotent, \( c_A(x) \) simplifies into the polynomial \( x^n \), which means that 0 is a root with a multiplicity equal to the size of the matrix.
• This tells us that \(\lambda = 0\) is an eigenvalue that occurs \( n \) times: one for each dimension of the square matrix.

This simplification is a direct result of the nilpotent property that \(A^m = 0\), emphasizing that only 0 contributes to the characteristic polynomial. With the matrix becoming entirely zero at a certain power, the factor-wise breakdown of the polynomial confirms \( x^n \) as the essence of the characteristic polynomial for a nilpotent matrix.