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Chapter 8: Problem 24

Suppose that \(\lambda\) is an eigenvalue of \(A .\) Show that \(\lambda^{2}\) is then an eigenvalue of \(A^{2}\) .

Short Answer

Expert verified
\( \lambda^2 \) is an eigenvalue of \( A^2 \) since \( A^2\mathbf{v} = \lambda^2\mathbf{v} \).

Step by step solution

01

Define Eigenvalue Equation

Suppose \( \mathbf{v} \) is a nonzero vector such that for matrix \( A \), the equation \( A\mathbf{v} = \lambda\mathbf{v} \) holds. By definition, \( \lambda \) is an eigenvalue of \( A \).
02

Apply Matrix \( A \) Again

To find if \( \lambda^2 \) is an eigenvalue of \( A^2 \), apply \( A \) once more to both sides of the eigenvalue equation: \( A(A\mathbf{v}) = A(\lambda\mathbf{v}) \).
03

Substitute and Derive

Substituting from the eigenvalue equation, we have \( A(\lambda\mathbf{v}) = \lambda(A\mathbf{v}) \). Then using \( A\mathbf{v} = \lambda\mathbf{v} \), this becomes \( \lambda(\lambda\mathbf{v}) = \lambda^2\mathbf{v} \).
04

Conclude Eigenvalue of \( A^2 \)

The equation \( A^2\mathbf{v} = \lambda^2\mathbf{v} \) shows that \( \lambda^2 \) acts on \( \mathbf{v} \) the same way as an eigenvalue does, confirming that \( \lambda^2 \) is indeed an eigenvalue of \( A^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eigenvectors
An eigenvector of a matrix is a special vector that maintains its direction after being transformed by the matrix.
In simpler terms, when you multiply a matrix by its eigenvector, the result is simply a scaled version of that eigenvector.
This scale factor is known as the eigenvalue. For instance, suppose we have a matrix \( A \) and a vector \( \mathbf{v} \), where \( A\mathbf{v} = \lambda\mathbf{v} \). Here, \( \mathbf{v} \) is the eigenvector, and the scaling factor \( \lambda \) is the eigenvalue.
  • Eigenvectors are not unique. If \( \mathbf{v} \) is an eigenvector, so is any non-zero scalar multiple of \( \mathbf{v} \).
  • Eigenvectors provide insight into the structure of a matrix and can reveal pivotal information about its properties.
Understanding eigenvectors is crucial as they help capture the directionality and fundamental aspects of matrices that are essential in fields like physics and engineering.
Matrix Algebra
Matrix algebra is the branch of mathematics that deals with matrices and their operations.
Matrices are arrays of numbers that allow us to perform complex linear algebra computations efficiently.
Some fundamental operations in matrix algebra include addition, subtraction, and multiplication of matrices.
  • Addition/Subtraction: This operation involves adding or subtracting corresponding elements of matrices with identical dimensions.
  • Multiplication: This operation is more complex. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. The product is a new matrix where each element is the sum of products of corresponding elements from the rows and columns.
  • Inversion: Not every matrix can be inverted, but for those that can, the inverse matrix \( A^{-1} \) satisfies \( AA^{-1} = I \), where \( I \) is the identity matrix.
Matrix algebra is foundational in many applications, including computer graphics, statistical modeling, and the study of network systems.
Linear Transformations
Linear transformations are functions that map vectors from one space to another while preserving operations of vector addition and scalar multiplication.
In mathematics, these transformations are often represented by matrices.
A transformation represented by a matrix \( A \) takes a vector \( \mathbf{v} \) in space and maps it to a new vector \( A\mathbf{v} \).
  • Preservation of operations: Linear transformations ensure that the addition of vectors and scalar multiplication behavior remains the same even after transformation.
  • Matrix Representation: Every linear transformation can be represented by a matrix, providing a concrete way to analyze and utilize these transformations efficiently.
  • Applications: They are widely used in computer graphics for scaling, rotating, and translating objects, as well as in robotics, physics, and economics.
By studying linear transformations, we gain a greater understanding of how spaces and functions interact, making it easier to manipulate and use them in practical applications.

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