Deflation: Suppose A has eigenvalue λand corresponding eigenvector 𝐯. (a) Let 𝐛 be any vector. P

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Deflation: Suppose A has eigenvalue λand corresponding...

Deflation: Suppose $A$ has eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) Let $\mathbf{b}$ be any vector. Prove that the matrix $B=A-\mathbf{v b}^T$ also has $\mathbf{v}$ as an eigenvector, now with eigenvalue $\lambda-\beta$, where $\beta=\mathbf{v}-\mathbf{b}$. (b) Prove that if $\mu \neq \lambda-\beta$ is any other eigenvalue of $A$, then it is also an eigenvalue of $B$. Hint: Look for an eigenvector of the form $\mathbf{w}+c \mathbf{v}$, where w is an eigenvector of $A$. (c) Given a nonsingular matrix $A$ with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ and $\lambda_1 \neq \lambda_j$ for all $j \geq 2$, explain how to construct a deflated matrix $B$ whose eigenvalues are $0, \lambda_2, \ldots, \lambda_n$. (d) Try out your method on the matrices $\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$ and $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$.

Deflation: Suppose $A$ has eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$. (a) Let $\mathbf{b}$ be any vector. Prove that the matrix $B=A-\mathbf{v b}^T$ also has $\mathbf{v}$ as an eigenvector, now with eigenvalue $\lambda-\beta$, where $\beta=\mathbf{v}-\mathbf{b}$. (b) Prove that if $\mu \neq \lambda-\beta$ is any other eigenvalue of $A$, then it is also an eigenvalue of $B$. Hint: Look for an eigenvector of the form $\mathbf{w}+c \mathbf{v}$, where w is an eigenvector of $A$. (c) Given a nonsingular matrix $A$ with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ and $\lambda_1 \neq \lambda_j$ for all $j \geq 2$, explain how to construct a deflated matrix $B$ whose eigenvalues are $0, \lambda_2, \ldots, \lambda_n$.
(d) Try out your method on the matrices $\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$ and $\left(\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right)$.

Key Concepts

Deflation Technique

The deflation technique is a method used in iterative eigenvalue algorithms to remove known eigenvalues from consideration. By modifying the original matrix (often via a rank-one update), the deflation process effectively 'deflates' or eliminates an eigenvalue, setting it to a predetermined value (such as zero) and thus simplifying the computation of the remaining eigenvalues. This is especially useful in large-scale problems where only a few dominant eigenvalues are sought.

Invariant Subspaces

An invariant subspace is a subspace that remains unchanged under the action of a linear operator. When an eigenvector belongs to such a subspace, modifications to the matrix (such as those introduced by a rank-one update in deflation) can be designed so that the eigenvector remains within this invariant subspace, preserving certain spectral properties. This idea is fundamental to understanding how deflation methods can be tailored to leave most of the eigensystem unaltered while adjusting a particular eigenvalue.

Eigenvalues and Eigenvectors

These are fundamental concepts in linear algebra where an eigenvalue of a matrix is a scalar that quantifies the factor by which the corresponding eigenvector is stretched during the linear transformation. An eigenvector is a nonzero vector that only changes by a scalar factor when that linear transformation is applied. Understanding this concept is crucial because it underpins many techniques in matrix analysis and numerical methods for solving systems of equations and differential equations.

Rank-One Update

A rank-one update involves modifying a matrix by adding or subtracting an outer product of two vectors. This type of update is significant because it alters the spectral properties of the matrix in a controlled way, often changing just one eigenvalue or leaving most eigenvalues intact. In the context of deflation, a rank-one update is used to selectively modify one eigenvalue while leaving the others almost unchanged.

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