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Chapter 6: Problem 10

Let \(A\) be an \(n \times n\) matrix and let \(B=A-\alpha I\) for some scalar \(\alpha .\) How do the eigenvalues of \(A\) and \(B\) compare? Explain.

Short Answer

Expert verified
In short, the eigenvalues of B are the eigenvalues of A shifted by 伪. If 位 is an eigenvalue of A, then 位 + 伪 is an eigenvalue of B, as can be observed from the relation \( \det(A - (\lambda + \alpha) I) = 0 \) for matrix B.

Step by step solution

01

Eigenvalues of A

To compute the eigenvalues of A, we will compute the determinant of (A - 位I), where 位 represents the eigenvalues and I is the identity matrix. \( \det(A - \lambda I) = 0 \)
02

Eigenvalues of B

Now, to compute the eigenvalues of B, we will compute the determinant of (B - 位I), where 位 represents the eigenvalues. Since B = A - 伪I, we can rewrite the determinant as follows: \( \det(B - \lambda I) = \det((A - \alpha I) - \lambda I) = \det(A - (\lambda + \alpha) I) = 0 \)
03

Compare eigenvalues of A and B

Now that we have the expressions for the eigenvalues of A and B, let's analyze the results. For A: \( \det(A - \lambda I) = 0 \) For B: \( \det(A - (\lambda + \alpha) I) = 0 \) We can see that the characteristic polynomial of B (matrix A - 伪I) is obtained by replacing 位 with (位 + 伪) in the characteristic polynomial of A. Therefore, if 位 鈭� spec(A), then 位 + 伪 鈭� spec(B). This means that the eigenvalues of B are the eigenvalues of A shifted by 伪.

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Key Concepts

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

Matrix Theory
Matrix theory is a key component of linear algebra, focusing on the study of matrices and their properties. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices can represent a wide variety of things including systems of linear equations, transformations, and graphs.

The most important notions in matrix theory include:
  • Matrix multiplication: A method used to combine two matrices to produce another matrix. Essential for transformations and operations in various fields.
  • Determinant: A scalar value that can be computed from the elements of a square matrix. It offers insights into matrix properties, including invertibility.
  • Identity matrix: A matrix that does not change another matrix when multiplied by it, similar to how multiplying by one does not change a number.
Understanding these elements is crucial for looking into eigenvalues and their shifts in matrix behaviours.
Linear Transformations
Linear transformations are functions that map one vector space to another, preserving vector addition and scalar multiplication. They play a vital role in mathematics, especially when dealing with matrices.

To comprehend linear transformations, consider these key points:
  • Transformation Matrices: Each linear transformation can be represented by a matrix that dictates how vectors in the original space are changed into the transformed space.
  • Invariant Subspaces: For certain linear transformations, some subspaces remain unchanged. These invariant subspaces are crucial when analyzing transformations and finding eigenvalues.
  • Relevance to Eigenvalues: In the context of eigenvalues, a transformation matrix can help identify how vectors stretch or rotate, allowing for deeper analysis of matrices such as calculating eigenvalue shifts.
Linear transformations underlie many applications in various fields, from computer graphics to quantum mechanics.
Characteristic Polynomial
The characteristic polynomial is a central tool when investigating eigenvalues of a matrix. It is derived from the matrix equation \(\det(A - \lambda I) = 0\), where \lambda\ is a scalar.

Here are some insights about characteristic polynomials:
  • Construction: The polynomial is formed by taking the determinant of \A - \lambda I\. Here, \A\ is the given matrix and \I\ is the identity matrix.
  • Eigenvalues: The solutions to the characteristic polynomial are the eigenvalues of the matrix, representing the factors by which a vector's length is extended or contracted without changing its direction.
  • Symmetric Matrices: For this special class of matrices, the characteristic polynomial has real roots, simplifying the calculation of eigenvalues.
The characteristic polynomial not only reveals eigenvalues but also helps in determining the matrix's stability and behavior.
Eigenvalue Shift
An eigenvalue shift occurs when a constant (scalar value) is added to a matrix, changing its eigenvalues by that constant. This concept is crucial in understanding how alterations to a matrix affect its properties.

To comprehend eigenvalue shifts, remember:
  • Algebraic Manipulation: If matrix \B\ is defined as \A - \alpha I\ (where \alpha\ is a scalar), then the eigenvalues of \B\ are the original eigenvalues of \A\, each decreased by \alpha\.
  • Equation Analysis: For matrix \A\, the characteristic equation is \(\det(A - \lambda I) = 0\). For matrix \B\, it becomes \(\det(A - (\lambda + \alpha) I) = 0\).
  • Practical Implications: Understanding eigenvalue shifts helps in scenarios where stability and control are critical, such as in engineering systems and physics simulations.
This shift offers valuable insights into the dynamic nature of systems modeled by matrices.

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Most popular questions from this chapter

Let \(A\) be a \(5 \times 5\) matrix with real entries. Let \(A=\) \(Q T Q^{T}\) be the real Schur decomposition of \(A,\) where \(T\) is a block matrix of the form given in equation \((2) .\) What are the possible block structures for \(T\) in each of the following cases? (a) All of the eigenvalues of \(A\) are real. (b) \(A\) has three real eigenvalues and two complex eigenvalues. (c) \(A\) has one real eigenvalue and four complex eigenvalues.

Find the eigenvalues of each of the following matrices and verify that conditions (i), (ii), and (iii) of Theorem 6.8 .1 hold: (a) \(\left(\begin{array}{ll}2 & 3 \\ 2 & 1\end{array}\right)\) (b) \(\left(\begin{array}{ll}4 & 2 \\ 2 & 7\end{array}\right)\) (c) \(\left(\begin{array}{lll}1 & 2 & 4 \\ 2 & 4 & 1 \\ 1 & 2 & 4\end{array}\right)\)

We can show that, for an \(n \times n\) stochastic matrix, \(\lambda_{1}=1\) is an eigenvalue and the remaining eigenvalues must satisfy \\[ \left|\lambda_{j}\right| \leq 1 \quad j=2, \ldots, n \\] (See Exercise 24 of Chapter \(7,\) Section \(4 .\) ) Show that if \(A\) is an \(n \times n\) stochastic matrix with the property that \(A^{k}\) is a positive matrix for some positive integer \(k\), then \\[ \left|\lambda_{j}\right|<1 \quad j=2, \ldots, n \\]

Let \(A\) be the PageRank transition matrix and let \(\mathbf{x}_{k}\) be a vector in the Markov chain with starting probability vector \(\mathbf{x}_{0} .\) since \(n\) is very large, the direct multiplication \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) is computationally intensive. However, the computation can be simplified dramatically if we take advantage of the structured components of \(A\) given in equation \((5) .\) Because \(M\) is sparse, the multiplication \(\mathbf{w}_{k}=M \mathbf{x}_{k}\) is computationally much simpler. Show that if we set \\[ \mathbf{b}=\frac{1-p}{n} \mathbf{e} \\] then \\[ E \mathbf{x}_{k}=\mathbf{e} \quad \text { and } \quad \mathbf{x}_{k+1}=p \mathbf{w}_{k}+\mathbf{b} \\] where \(M, E, \mathbf{e},\) and \(p\) are as defined in equation (5)

A management student received fellowship offers from four universities and now must choose which one to accept. The student uses the analytic hierarchy process to decide among the universities and bases the decision process on the following four criteria: (i) financial matters - tuition and scholarships (ii) the reputation of the university (iii) social life at the university (iv) geography-how desirable is the location of the university In order to weigh the criteria the student decides that finance and reputation are equally important and both are 4 times as important as social life and 6 times as important as geography. The student also rates social life twice as important as geography. (a) Determine a reciprocal comparison matrix \(C\) based on the given judgments of the relative importance of the 4 criteria. (b) Show that the matrix \(C\) is not consistent. (c) Make the problem consistent by changing the relative importance of one pair of criteria and determine a new comparison matrix \(C_{1}\) for the consistent problem. (d) Find an eigenvector belonging to the dominant eigenvalue of \(C_{1}\) and use it to determine a weight vector for the decision criteria.
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