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Chapter 5: Problem 36

Let \(A\) be an \(n \times n\) matrix and let \(D_{1}, D_{2}, \ldots, D_{n}\) be its Gershgorin disks. Suppose that an eigenvalue \(\lambda\) lies in \(D_{k}\) but not in any of the other disks. Let \(x\) be an eigenvector corresponding to \(\lambda\). Show that \(\left|x_{k}\right|>\left|x_{i}\right|\) for \(i \neq k\).

Short Answer

Expert verified
The eigenvector component \(x_k\) corresponding to the isolated disk is the largest.

Step by step solution

01

Understanding the Gershgorin Disk Theorem

The Gershgorin Disk Theorem states that every eigenvalue of matrix \(A\) lies within at least one of the Gershgorin disks \(D_k\). Each disk \(D_k\) is centered at the diagonal entry \(a_{kk}\) with radius \(R_k = \sum_{i eq k} |a_{ki}|\).
02

Identifying the Isolated Disk

It is given that an eigenvalue \(\lambda\) of \(A\) lies within the disk \(D_k\) and not in any other disk. This suggests that \(D_k\) is an isolated disk containing \(\lambda\).
03

Eigenvalue-Eigenvector Relationship

Let \(x\) be an eigenvector corresponding to \(\lambda\), such that \(Ax = \lambda x\). If \(A = [a_{ij}]\), then the equation for each component \(x_i\) is \(\sum_{j=1}^{n} a_{ij} x_j = \lambda x_i\).
04

Analyzing the Gershgorin Disk

Since \(\lambda\) lies in \(D_k\), it satisfies the inequality \(|a_{kk} - \lambda| < R_k\). For every disk \(D_i\) with \(i eq k\), the eigenvalue \(\lambda\) does not satisfy this inequality, meaning \(|a_{ii} - \lambda| \geq R_i\).
05

Comparing Eigenvector Components

From the equation \(\sum_{j=1}^{n} a_{kj} x_j = \lambda x_k\), isolating \(x_k\) we have \(a_{kk} x_k + \sum_{j eq k} a_{kj} x_j = \lambda x_k\). Since \(\lambda \) is closer to \(a_{kk}\) than the sum of the off-diagonal terms, \(x_k\) must be the dominant term. This implies \(|x_k| > |x_i|\) for \(i eq k\).
06

Conclusion

Thus, because \(\lambda\) lies only in disk \(D_k\), the component \(x_k\) of the eigenvector \(x\) is larger than other components.

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Key Concepts

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

Eigenvalues
Eigenvalues are critical to understanding the behavior of a matrix. An eigenvalue, denoted as \(\lambda\), is a scalar that measures how a matrix scales a vector. More formally, for a given square matrix \(A\), a non-zero vector \(x\) is an eigenvector associated with \(\lambda\) if the equation \(Ax = \lambda x\) holds true. This means when \(A\) acts on \(x\), \(x\) only scales or stretches by \(\lambda\) without changing direction.
  • Each eigenvalue has corresponding eigenvectors, potentially forming a span.
  • Finding these values involves solving the characteristic equation \(\det(A - \lambda I) = 0\).
  • Eigenvalues are important in determining the matrix's properties, such as stability in systems and physical applications.
A deeper understanding of eigenvalues is enhanced through concepts like the Gershgorin Disk Theorem. This theorem aids in locating eigenvalues by using disks plotted in the complex plane, centered on diagonal elements of the matrix.
Eigenvectors
Eigenvectors are vectors that maintain their direction when a transformation is applied through a matrix. These vectors, corresponding to eigenvalues, describe in which directions the transformation via the matrix \(A\) scales the space. For a matrix, while there are multiple eigenvectors associated with each eigenvalue \(\lambda\), they all lie on the same line passing through the origin.
  • Eigenvectors can be scaled; if \(x\) is an eigenvector, so is any scalar multiple \(cx\), where \(c\) is a non-zero scalar.
  • The set of all eigenvectors associated with a particular eigenvalue plus the zero vector forms a subspace, known as the eigenspace.
  • In practical terms, eigenvectors denote the principle axes in transformations or simplifying complex systems.
In our exercise, when an eigenvalue \(\lambda\) uniquely lies in a Gershgorin disk, the corresponding eigenvector has a distinct property: one of its components, \(x_k\), is significantly larger.
Matrix Analysis
Matrix analysis explores the properties and functionalities of matrices, providing tools for advanced mathematical applications. It often involves studying eigenvalues and eigenvectors, offering insights into systems modeled by matrices.
  • Matrix analysis allows simplification of complex systems, finding simplified forms like the diagonal or Jordan forms.
  • The study helps determine matrix stability and the behavior of linear transformations.
  • It assists in decomposing matrices, making them manageable for computations and solutions of differential equations.
  • Key concepts: determinants, norms, ranks, and special matrix types like symmetric or Hermitian.
Using tools like the Gershgorin Disk Theorem, matrix analysis can localize eigenvalues within specific regions of the complex plane, thus providing bounds and approximations for the behavior of the matrix \(A\). Understanding disks and their isolation properties explicates conditions on components of eigenvectors, which is central to leveraging eigenstuff in practical applications.

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Most popular questions from this chapter

Which elementary row operations on the system \(A x=b\) leave all least-squares solutions unchanged?

Determine an approximate value for the spectral radius, \(\rho(A)\), of the following matrix by taking two iterations of the power method using the infinity norm for \(\varphi .\) Use the initial vector \((1,1,1)^{T} .\) (A norm is not a linear functional.) $$ A=\left[\begin{array}{rrr} 2 & 0 & -1 \\ -2 & -10 & 0 \\ -1 & -1 & 4 \end{array}\right] $$

Find the minimal solutions of these systems of equations: a. \(x_{1}+x_{2}=b_{1}\) b. \(\left\\{\begin{array}{l}x_{1}=b_{1} \\ x_{1}=b_{2} \\\ x_{1}=b_{3}\end{array}\right.\) c. \(\left\\{\begin{array}{l}4 x_{1}=b_{1} \\ 0 x_{1}=b_{2} \\ 7 x_{3}=b_{3} \\\ 0 x_{2}=b_{4}\end{array}\right.\)

Suppose that \(A=U D V\), where \(U\) is \(m \times m\) unitary, \(V\) is \(n \times n\) unitary, and \(D\) is \(m \times n\) diagonal. Show that \(\left|d_{i i}\right|^{2}(1 \leq i \leq n)\) are the eigenvalues of \(A^{*} A\).

If \(A\) is Hermitian and positive definite, what is its singular-value decomposition?
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