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Chapter 8: Problem 8

Suppose that $$ A=\left[\begin{array}{cc} D & v \\ v^{T} & d_{n} \end{array}\right] $$ where \(D=\operatorname{diag}\left(d_{1}, \ldots, d_{n-1}\right)\) has distinct diagonal entries and \(v \in \mathbf{R}^{n-1}\) has no zero entries. (a) Show that if \(\lambda \in \lambda(A)\), then \(D-\lambda I_{n-1}\) is nonsingular. (b) Show that if \(\lambda \in \lambda(A)\), then \(\lambda\) is a zero of $$ f(\lambda)=\lambda+\sum_{k=1}^{n-1} \frac{v_{k}^{2}}{d_{k}-\lambda}-d_{n} $$

Short Answer

Expert verified
(a) \(D - \lambda I_{n-1}\) is nonsingular. (b) \(\lambda\) is a zero of \(f(\lambda).\)

Step by step solution

01

Introduction

We need to solve two parts: (a) Show that \(D - \lambda I_{n-1}\) is nonsingular if \(\lambda\) is an eigenvalue of \(A\). (b) Show \(\lambda\) is a zero of the given function.
02

Understanding the Structure of A

\(A\) is a block matrix composed of a diagonal matrix \(D\) and vectors \(v\) and \(v^T\). So, \(A = \begin{bmatrix} D & v \ v^T & d_n \end{bmatrix}\), where \(D = \text{diag}(d_1, \ldots, d_{n-1})\).
03

Determinant Approach for (a)

The eigenvalue \(\lambda\) makes \(A - \lambda I\) singular, so its determinant is zero: \(\det(A - \lambda I) = 0\). For block matrices, \(A - \lambda I = \begin{bmatrix} D - \lambda I_{n-1} & v \ v^T & d_n - \lambda \end{bmatrix}\).
04

Using Schur's Complement for (a)

The determinant of a block matrix is \(\det(D - \lambda I_{n-1}) \cdot ((d_n - \lambda) - v^T (D - \lambda I_{n-1})^{-1} v)\). If \(\lambda\) makes \(A - \lambda I\) singular, then \(D - \lambda I_{n-1}\) must be nonsingular to avoid a zero product.
05

Function Setup for (b)

We need to show \(\lambda\) is a zero of: \(f(\lambda) = \lambda + \sum_{k=1}^{n-1} \frac{v_k^2}{d_k - \lambda} - d_n\). Since this follows the eigenvalue condition, setup also considers the effect of \((d_n - \lambda) - v^T (D - \lambda I_{n-1})^{-1} v=0\), simplify this equation.
06

Investigate Zero Condition in (b)

Rewriting \((d_n - \lambda) = v^T (D - \lambda I_{n-1})^{-1} v\) equivalent to the root condition, imply this sum equals \(d_n\). Substitute back into \(\lambda\) equation.
07

Equality and Conclusion for (b)

Thus \(\lambda\) satisfies \(\lambda + \sum_{k=1}^{n-1} \frac{v_k^2}{d_k - \lambda} = d_n\), this confirms \(f(\lambda)\) zeroes out due to eigenvalue conditions.

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Key Concepts

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

Block Matrices
Block matrices are matrices that are divided into smaller matrices, or "blocks." This structure can be particularly useful in simplifying complex computations when dealing with large matrices. Consider a block matrix \(A\) of the form:\[A = \begin{bmatrix} D & v \ v^T & d_n \end{bmatrix}\]Here, \(D\) is a diagonal matrix and \(v\) is a column vector.
  • Block matrices allow operations like inversion and determinant calculations to be broken down into operations on smaller blocks.
  • This is especially beneficial when the blocks have properties like being diagonal or easily invertible, which makes operations more efficient.
Working with block matrices is common in areas such as control theory and optimization as they can simplify the representation and manipulation of large systems.
Diagonal Matrix
A diagonal matrix is a special type of square matrix where all off-diagonal elements are zero. Only the entries on the main diagonal can have non-zero values, which gives the matrix its "diagonal" property. The matrix \(D\) in our block matrix A is a diagonal matrix represented as:\[D = \operatorname{diag}(d_1, d_2, \ldots, d_{n-1})\]
  • Diagonal matrices are important because they are the simplest type of matrices to understand and work with.
  • They are automatically invertible if none of the diagonal entries are zero.
  • Calculations like finding determinants and eigenvalues are straightforward: the determinant is simply the product of the diagonal elements.
In the context of eigenvalue problems, diagonal matrices serve as a convenient simplification, making various mathematical operations and derivations more manageable.
Schur's Complement
Schur's Complement is a powerful mathematical tool used to simplify the process of calculating the determinant of a block matrix. In essence, it provides a pathway to transform complex block matrix problems into more familiar and manageable forms. For our block matrix \(A\), the determinant using Schur's Complement is given by:\[\det(A - \lambda I) = \det(D - \lambda I_{n-1}) \cdot \left((d_n - \lambda) - v^T (D - \lambda I_{n-1})^{-1} v\right)\]
  • The expression \((d_n - \lambda) - v^T (D - \lambda I_{n-1})^{-1} v\) is known as the Schur's Complement in this scenario.
  • This determinant condition helps analyze systems of linear equations, particularly in establishing whether a matrix is singular or nonsingular.
  • Understanding Schur's Complement is crucial for insight into stability and control of systems.
By utilizing Schur's Complement, the properties of the larger matrix \(A\) can be inferred by understanding the properties of its individual blocks, especially when an eigenvalue approach is needed.
Determinant
The determinant is a scalar value that encapsulates a matrix's key properties, revealing crucial information about the matrix. It鈥檚 a measure of a matrix鈥檚 "volume" expansion factor in linear transformations. When dealing with the matrix \(A\), calculating its determinant is imperative for understanding if \(A - \lambda I\) is singular:\[\det(A - \lambda I) = 0\]
  • A determinant of zero indicates that the matrix is singular, meaning it does not have full rank and cannot be inverted.
  • Determinants play a role in solving systems of linear equations, particularly in determining the existence and uniqueness of solutions.
  • They are also fundamentally used in deriving characteristic polynomials, which are essential in finding eigenvalues.
By understanding the determinant, one gains significant insight into the solvability and stability of systems that the matrix seeks to represent.

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Most popular questions from this chapter

Let $$ A=\left[\begin{array}{ll} w & x \\ x & z \end{array}\right] $$ be real and suppose we perform the following shifted QR step: \(A-z I=U R, \vec{A}=R U+z I\). Show that $$ \hat{A}=\left[\begin{array}{rr} \hat{w} & \bar{x} \\ \hat{x} & z \end{array}\right] $$ where $$ \begin{aligned} &\bar{w}=w+x^{2}(w-z) /\left[(w-z)^{2}+x^{2}\right] \\ &\bar{z}=z-x^{2}(w-z) /\left[(w-z)^{2}+x^{2}\right] \\ &\vec{x}=-x^{3} /\left[(w-z)^{2}+x^{2}\right] \end{aligned} $$

Let \(A \in \mathbf{R}^{n \times n}\) be symmetric. Give an algorithm that computes the factorization $$ Q^{T} A Q=\gamma I+F $$ where \(Q\) is a product of Jacobi rotations, \(\gamma=\operatorname{tr}(A) / n\), and \(F\) has zero diagonal entries. Discuss the uniqueness of \(Q\).

Suppose \(A \in \mathbb{C}^{n \times n}\) is Hermitian. Show how to construct unitary \(Q\) such that \(Q^{H} A Q=T\) is real, symmetric, and tridiagonal.

Show that the eigenvalues of a Hermitian matrix \(\left(A^{H}=A\right)\) are real. For each theorem and corollary in this section, state and prove the corresponding result for Hermitian matrices. Which results have analogs when \(A\) is skew-symmetric? Hint: If \(A^{T}=-A\), then \(i A\) is Hermitian.

Assume that \(n=2 m\) and that \(S \in \mathbf{R}^{n \times n}\) is skew-symmetric and tridiagonal. Show that there exists a permutation \(P \in \mathbf{R}^{n \times n}\) such that $$ P^{T} S P=\left[\begin{array}{cc} 0 & -B^{T} \\ B & 0 \end{array}\right] $$ where \(B \in \mathbf{R}^{m \times m}\). Describe the structure of \(B\) and show how to compute the eigenvalues and eigenvectors of \(S\) via the SVD of \(B\). Repeat for the case \(n=2 m+1\).
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