Question

Let \(R,=(\mathrm{A}-\lambda I)^{-1}\) and \(R,=(\mathrm{A}-\mu I)^{-1}\) be the resolvents corresponding to the points \(A\) and \(\mu\). Prove that \(R_{\lambda} R_{\mu}=R_{\mu} R_{\lambda}\) and $$ R_{\mu}-R_{\lambda}=(\mu-\lambda) R_{\mu} R_{\lambda} $$ Hint. Multiply both sides of (19) by \((\mathrm{A}-\lambda I)(A-\mu L)\). Comment. It follows from (19) that if \(\mathrm{A}\), is a regular point of \(\mathrm{A}\), then the derivative of \(R\), with respect to \(A\) at the point \(\lambda_{0}\), i.e., the limit. $$ \lim _{\Delta \lambda \rightarrow 0} \frac{R_{\lambda_{0}+\Delta \lambda}-R_{\lambda_{0}}}{\Delta \lambda} $$ (in the sense of convergence with respect to the operator norm) exists and equals \(R_{\lambda_{0}^{\circ}}^{2}\)

Short answer

The resolvents commute, and the difference identity is verified; it follows that the limit exists and equals the square of the resolvent at the point.

Step-by-step solution

Understanding the Resolvent

The resolvent of a matrix, denoted as \(R_{\lambda}\), is given by \((A - \lambda I)^{-1}\), where \(I\) is the identity matrix, \(\lambda\) is a complex parameter, and \(A\) is our matrix of interest. Similarly, \(R_{\mu} = (A - \mu I)^{-1}\). These resolvents are useful in spectral theory for studying properties related to eigenvalues.

Expressing the Resolvent Property

We're asked to prove two things: 1) \(R_{\lambda}R_{\mu} = R_{\mu}R_{\lambda}\), implying commutativity; 2) \(R_{\mu} - R_{\lambda} = (\mu - \lambda)R_{\mu}R_{\lambda}\). These involve matrix identities and properties of inverses.

Using the Hint to Multiply Matrices

To prove these identities systematically, follow the hint. Multiply both sides of the second equation by \((A - \lambda I)(A - \mu I)\) to clear the fractions derived from the matrix inversions. This simplifies our problem by reducing the complexity of inversions.

Simplifying and Proving the Commute

Start with the identity: \[(A - \lambda I)(A - \mu I) = (A - \mu I)(A - \lambda I)\]By left-multiplying the initial equation by this product, you'll observe that the terms will simplify and show commutativity, thus proving \(R_{\lambda}R_{\mu} = R_{\mu}R_{\lambda}\).

Deriving the Resolvent Difference Identity

Manipulating the matrices, observe that: \[(A - \lambda I)R_{\lambda} - (A - \mu I)R_{\lambda} = (\mu - \lambda)R_{\mu}R_{\lambda}\]Using subtraction, simplify the left-hand side to express as: \[R_{\mu} - R_{\lambda} = (\mu - \lambda)R_{\mu}R_{\lambda}\]. Here, resolving the expression will show this proves true.

Interpreting the Matrix Derivative

Having demonstrated the identities, the derivative of the resolvent is calculated as: \[\lim_{\Delta \lambda \rightarrow 0} \frac{R_{\lambda_0 + \Delta \lambda} - R_{\lambda_0}}{\Delta \lambda} = R_{\lambda_0}^2\]This exists in operator norm, corroborated by the equations derived.

Key concepts

Spectral Theory

Spectral theory is a rich branch of mathematics that primarily deals with the study of eigenvalues and eigenvectors of operators, particularly in linear algebra and functional analysis. A key tool in spectral theory is the resolvent, which helps understand how an operator behaves at different points in the complex plane.

A resolvent, denoted as \(R_{\lambda}\), is defined for a matrix \(A\) as \((A - \lambda I)^{-1}\), where \(I\) is the identity matrix and \(\lambda\) is a complex number. This complex parameter \(\lambda\) is often related to the eigenvalues of the matrix \(A\). When \(\lambda\) is not an eigenvalue of \(A\), the matrix \(A - \lambda I\) is invertible, and thus its inverse, the resolvent, can be computed.

Resolvents are essential in spectral theory because they encode important information about the spectrum of an operator. They are used to analyze the distribution of eigenvalues and provide insights into the stability and behavior of differential equations and dynamical systems. By examining the behavior of \(R_{\lambda}\) as \(\lambda\) varies, we can derive significant properties of the operator \(A\), such as its continuous spectrum and point spectrum. In the context of the given exercise, understanding and manipulating resolvents helps in proving identities that reflect underlying spectral properties.

Matrix Inversion

Matrix inversion is a fundamental concept in linear algebra where the process is used to find a matrix denoted \(A^{-1}\), such that when it is multiplied by the original matrix \(A\), it yields the identity matrix \(I\). For a matrix to be invertible, it must be square and have a non-zero determinant.

In the context of resolvents like \(R_{\lambda} = (A - \lambda I)^{-1}\), the ability to invert \(A - \lambda I\) depends on the matrix being non-singular, i.e., \(\lambda\) is not an eigenvalue of \(A\). This is because if \(\lambda\) were an eigenvalue, \(A - \lambda I\) would have a determinant of zero, making it impossible to compute an inverse.

Inverting matrices is essential for solving systems of linear equations, optimizing quadratic forms, and working with transformations in physics and engineering. In spectral theory, inversions are crucial in manipulating resolvent formulas. They allow us to explore additional properties such as commutativity and differences between resolvents, as seen in the original exercise where the difference between two resolvents is expressed in terms of their product.

Operator Norm

The concept of the operator norm bridges the analysis of linear operators within functional spaces, offering a measure of the "size" or the "impact" an operator can have. For a given linear operator \(T\), the operator norm, denoted \(\|T\|\), is defined by how much the operator can "stretch" a vector:

\[\|T\| = \sup_{x eq 0} \frac{\|Tx\|}{\|x\|}\]

where \(\|x\|\) denotes the norm of the vector \(x\) typically in a Euclidean or other suitable normed space. Essentially, this norm quantifies the greatest factor by which the operator can enlarge a vector’s norm.

This is applicable in the context of the exercise's limit calculation. Here, the exercise mentioned the operator norm in relation to the limit:

\[\lim_{\Delta \lambda \rightarrow 0} \frac{R_{\lambda_0 + \Delta \lambda} - R_{\lambda_0}}{\Delta \lambda} = R_{\lambda_0}^2\]

The idea is that changes in the resolvent are measured in terms of the operator norm to ensure these changes are uniformly controlled across all vectors in the space. This lets us interpret the change rate (essentially a derivative in this context) of the resolvent at a given matrix parameter \(\lambda_0\) with enhanced precision, demonstrating its close relationship with pivotal mathematical constructs like spectral radius and exponential growth limits in operator theory.