Question

Show that if \(\mathbf{A}\) is invertible, then the eigenvectors of \(\mathbf{A}^{-1}\) are the same as those of \(\mathbf{A}\) and the eigenvalues of \(\mathbf{A}^{-1}\) are the reciprocals of those of \(\mathbf{A}\).

Short answer

The eigenvectors of an invertible matrix and its inverse are the same. The eigenvalues of the inverse matrix are the reciprocals of the ones of the original matrix.

Step-by-step solution

Define what is given

An invertible matrix \(\mathbf{A}\) is given. This means there exists a matrix \(\mathbf{A}^{-1}\) such that \(\mathbf{A} \mathbf{A}^{-1} = \mathbf{A}^{-1} \mathbf{A} = \mathbf{I}\). Also, it is known that \(\mathbf{A}\) has eigenvectors and eigenvalues as per the definition, \(\mathbf{A} \mathbf{x} = \lambda \mathbf{x}\), where \(\mathbf{x}\) is the eigenvector and \(\lambda\) is the corresponding eigenvalue.

Show the eigenvectors are the same

To prove the eigenvectors are the same, multiply the eigenvalue equation by \(\mathbf{A}^{-1}\). This gives \(\mathbf{A}^{-1} \mathbf{A} \mathbf{x} = \lambda \mathbf{A}^{-1} \mathbf{x}\), which simplifies to \(\mathbf{x} = \lambda \mathbf{A}^{-1} \mathbf{x}\) concluding that \(\mathbf{x}\) is also an eigenvector of \(\mathbf{A}^{-1}\). So, the eigenvectors of \(\mathbf{A}^{-1}\) are the same as those of \(\mathbf{A}\).

Show the eigenvalues are reciprocals

To find the corresponding eigenvalue for \(\mathbf{A}^{-1}\), rewrite the above equation as \(\mathbf{A}^{-1} \mathbf{x} = \frac{1}{\lambda} \mathbf{x}\), which suggests that the eigenvalues of \(\mathbf{A}^{-1}\) are the reciprocals of those of \(\mathbf{A}\).

Key concepts

Invertible Matrix

In linear algebra, an invertible matrix, also known as a non-singular matrix, is a square matrix that possesses a unique inverse.

For a matrix \( \mathbf{A} \) to be invertible, there must exist another matrix \( \mathbf{A}^{-1} \) such that their product in both orders—\( \mathbf{A} \mathbf{A}^{-1} \) and \( \mathbf{A}^{-1} \mathbf{A} \)—yields the identity matrix \( \mathbf{I} \). The identity matrix is a special kind of diagonal matrix with ones on the diagonal and zeros elsewhere, and it acts as the multiplicative identity for matrices, meaning any matrix multiplied by it remains unchanged.

An important property of invertible matrices connected to eigenvectors and eigenvalues is that transforming an eigenvector by an invertible matrix results in a scaling of the eigenvector, not a change in direction. This quality maintains the definition of an eigenvector after inversion, which is pivotal to understanding why the exercise solution is valid.

Reciprocal Eigenvalues

Eigenvalues are scalar values that, when multiplied by their associated eigenvectors, yield the same result as a given matrix transformation. The connection between an invertible matrix \( \mathbf{A} \) and its inverse \( \mathbf{A}^{-1} \) extends to their eigenvalues.

When \( \mathbf{A} \) is invertible, the reciprocal eigenvalues property emerges. This states that the eigenvalues of \( \mathbf{A}^{-1} \) are the reciprocals of \( \mathbf{A} \)’s eigenvalues. Mathematically, if \( \lambda \) is an eigenvalue of \( \mathbf{A} \) with eigenvector \( \mathbf{x} \), then \( \frac{1}{\lambda} \) (or \( \lambda^{-1} \) to signify a reciprocal) is an eigenvalue of \( \mathbf{A}^{-1} \).

This concept is not only fascinating but extremely useful because it allows us to compute the eigenvalues of \( \mathbf{A}^{-1} \) effortlessly from those of \( \mathbf{A} \) without conducting laborious matrix calculations. It's a succinct illustration of the intertwining relationships ubiquitous in linear algebra.

Linear Algebra

At its core, linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also fundamental to modern presentations of geometry, including for defining objects such as lines, planes, and rotations.

In the context of the given exercise, linear algebra provides the framework for understanding concepts like matrices, eigenvectors, and eigenvalues, which are crucial in various applications across engineering, physics, computer science, and more. The inner workings of systems, transformations, and rotations in multidimensional spaces are often explained using linear algebraic terms.

Mastering the basics of this discipline is key to advancing in any field involving mathematics or computational algorithms. It plays a pivotal role in the development of computer algorithms and is instrumental in machine learning, data mining, scientific simulations, and even in the study of social networks.