Question

Suppose \(\lambda\) is an eigenvalue of a square matrix \(A\) with eigenvector \(\mathbf{x} \neq \mathbf{0}\) a. Show that \(\lambda^{2}\) is an eigenvalue of \(A^{2}\) (with the same \(\mathbf{x})\) b. Show that \(\lambda^{3}-2 \lambda+3\) is an eigenvalue of \(A^{3}-2 A+3 I\) c. Show that \(p(\lambda)\) is an eigenvalue of \(p(A)\) for any nonzero polynomial \(p(x)\).

Short answer

\( \lambda^2 \), \( \lambda^3 - 2\lambda + 3 \), and \( p(\lambda) \) are eigenvalues of \( A^2 \), \( A^3 - 2A + 3I \), and \( p(A) \), respectively.

Step-by-step solution

Understand Eigenvalue Definition

By definition, if \( \lambda \) is an eigenvalue of a matrix \( A \) with eigenvector \( \mathbf{x} \), then \( A \mathbf{x} = \lambda \mathbf{x} \). This equation will be the basis for proving all parts of the problem.

Show \( \lambda^2 \) as an Eigenvalue of \( A^2 \)

To prove that \( \lambda^2 \) is an eigenvalue of \( A^2 \), apply matrix \( A \) to the eigenvector equation two times. Start with \( A \mathbf{x} = \lambda \mathbf{x} \), then apply \( A \) again: \( A(A \mathbf{x}) = A(\lambda \mathbf{x}) = \lambda A \mathbf{x} = \lambda(\lambda \mathbf{x}) = \lambda^2 \mathbf{x} \). Thus, \( A^2 \mathbf{x} = \lambda^2 \mathbf{x} \), proving \( \lambda^2 \) is an eigenvalue of \( A^2 \).

Show Expression as Eigenvalue of Operator on A

For \( \lambda^3 - 2 \lambda + 3 \), start by applying \( A \) three times to the eigenvector equation: \( A^2 \mathbf{x} = \lambda^2 \mathbf{x} \) and \( A^3 \mathbf{x} = (A^2)(A \mathbf{x}) = \lambda^3 \mathbf{x} \). Now consider \((A^3 - 2A + 3I) \mathbf{x} = A^3 \mathbf{x} - 2A \mathbf{x} + 3I \mathbf{x} = \lambda^3 \mathbf{x} - 2\lambda \mathbf{x} + 3 \mathbf{x} \). Factor out \( \mathbf{x} \): \( = (\lambda^3 - 2\lambda + 3) \mathbf{x} \), thus proving \( \lambda^3 - 2\lambda + 3 \) is an eigenvalue of \( A^3 - 2A + 3I \).

Generalize to Any Polynomial \( p(x) \)

Suppose \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \). The polynomial of the matrix \( A \) is \( p(A) = a_n A^n + a_{n-1} A^{n-1} + \dots + a_1 A + a_0 I \). When applied to \( \mathbf{x} \), each term \( a_i A^i \mathbf{x} = a_i \lambda^i \mathbf{x} \), thus \( p(A) \mathbf{x} = (a_n \lambda^n + a_{n-1} \lambda^{n-1} + \dots + a_1 \lambda + a_0) \mathbf{x} = p(\lambda) \mathbf{x} \). Therefore, \( p(\lambda) \) is an eigenvalue of \( p(A) \).

Key concepts

Matrix Polynomials

In mathematics, a matrix polynomial involves a polynomial expression where the variables are replaced by matrices. Matrix polynomials play a significant role in linear algebra, particularly when dealing with eigenvalues and eigenvectors. To understand matrix polynomials, begin with a regular polynomial expressed as:

• \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)

In this context, a matrix polynomial replaces the variable \( x \) with a square matrix \( A \), resulting in:

• \( p(A) = a_n A^n + a_{n-1} A^{n-1} + \ldots + a_1 A + a_0 I \)

Where \( I \) is the identity matrix of the same dimensions as \( A \). Applying polynomials to matrices allows us to explore deeper properties of linear transformations. Notably, if \( \lambda \) is an eigenvalue of \( A \), then for any polynomial \( p(x) \), \( p(\lambda) \) becomes an eigenvalue of \( p(A) \). This principle is immensely beneficial in solving complex systems and understanding matrix behaviors.

Eigenvectors

An eigenvector is a special vector that remains parallel to itself after a linear transformation represented by a matrix. When a square matrix \( A \) acts on an eigenvector \( \mathbf{x} \), the output is a scalar multiple of \( \mathbf{x} \), known as an eigenvalue \( \lambda \). The relationship is defined as:

• \( A \mathbf{x} = \lambda \mathbf{x} \)

Eigenvectors provide a lens through which to understand the intrinsic properties of matrices. While eigenvalues tell us about the magnitude of stretching or shrinking, eigenvectors indicate the directions along which this transformation occurs.
To find eigenvectors, one typically solves the equation \( (A - \lambda I) \mathbf{x} = \mathbf{0} \), where \( I \) is the identity matrix. The solutions to this equation correspond to the eigenvectors of \( A \), which are pivotal for tasks like diagonalization and stability analysis of systems.

Square Matrices

Square matrices are matrices with an equal number of rows and columns, forming a fundamental building block in linear algebra. These matrices are particularly interesting because they enable operations like determining eigenvalues and eigenvectors, matrix inverses, and computing determinants.
Some of the key properties and uses of square matrices include:

• **Determinants and Inversion:** A square matrix can have a determinant, which is a scalar value encoding important properties of the matrix, such as invertibility.
• **Eigenvalues and Eigenvectors:** As discussed earlier, square matrices can possess eigenvalues and eigenvectors that provide insights into the matrix's behavior under various transformations.
• **Diagonalization:** If a square matrix can be decomposed into a product of three matrices, it is diagonalizable, making it easier to compute matrix powers and exponentiate.

Square matrices form the backbone of many algorithms and operations in mathematics and engineering, making them essential to every student's toolkit.