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Chapter 6: Problem 7

Let \(A\) be an \(n \times n\) matrix and let \(B=I-2 A+A^{2}\) (a) Show that if \(\mathbf{x}\) is an eigenvector of \(A\) belonging to an eigenvalue \(\lambda\), then \(\mathbf{x}\) is also an eigenvector of \(B\) belonging to an eigenvalue \(\mu\) of \(B\). How are \(\lambda\) and \(\mu\) related? (b) Show that if \(\lambda=1\) is an eigenvalue of \(A\), then the matrix \(B\) will be singular

Short Answer

Expert verified
(a) If \(\mathbf{x}\) is an eigenvector of A with eigenvalue \(\lambda\), then \(A\mathbf{x} = \lambda\mathbf{x}\). We show that \(\mathbf{x}\) is also an eigenvector of B by finding the eigenvector equation for B: \(B\mathbf{x} = (I - 2A + A^2)\mathbf{x}\). After simplifying, we find that \(B\mathbf{x} = \mathbf{x}(1 - 2\lambda + \lambda^2)\). Thus, \(\mathbf{x}\) is an eigenvector of B with eigenvalue \(\mu = 1 - 2\lambda + \lambda^2\). (b) If \(\lambda = 1\) is an eigenvalue of A, then the eigenvalue of B is \(\mu = 1 - 2(1) + (1)^2 = 0\). Since B has an eigenvalue of 0, B is singular.

Step by step solution

01

Eigenvalue and eigenvector equation for A matrix

We know that a vector x is an eigenvector of A if it satisfies the following equation: \(A\mathbf{x} = \lambda\mathbf{x}\), where \(\lambda\) is an eigenvalue of A. Step 2: Replace A by its eigenvector expression in the B matrix
02

Finding eigenvector equation for B matrix

We want to show that x is also an eigenvector for B. To do this, we'll first find an equation for Bx: \(B\mathbf{x} = (I - 2A + A^2)\mathbf{x}\) Step 3: Plug in A eigenvector expression and simplify
03

Simplifying eigenvector equation for B

Let's plug in the A eigenvector expression and simplify the equation: \(B\mathbf{x} = (I\mathbf{x} - 2(\lambda\mathbf{x}) + A(A\mathbf{x}))\) Since A satisfies the eigenvector condition, we can plug in A and simplify: \(B\mathbf{x} = (\mathbf{x} - 2\lambda\mathbf{x} + A(\lambda\mathbf{x}))\) Step 4: Show x is an eigenvector for B and find the eigenvalue
04

Eigenvalue and eigenvector relationship

Now we can factor out the eigenvector x: \(B\mathbf{x} = \mathbf{x}(1 - 2\lambda + \lambda^2)\) This shows that x is also an eigenvector for B and the eigenvalue of B is related to the eigenvalue of A as follows: \(\mu = 1 - 2\lambda + \lambda^2\) (b) Step 1: Assume eigenvalue of A is 1
05

Assumption of eigenvalue for A

We are given that if \(\lambda = 1\) is an eigenvalue of A, then we want to show that B is singular. Step 2: Calculate the eigenvalue of B for \(\lambda = 1\)
06

Eigenvalue of B for given \(\lambda\)

If \(\lambda = 1\), we can plug it into our related eigenvalue expression found in part (a) to find the eigenvalue of B: \(\mu = 1 - 2(1) + (1)^2 = 0\) Step 3: Show that eigenvalue 0 implies B is singular
07

Singular matrix B from eigenvalue

We have found that if \(\lambda = 1\), then the eigenvalue of B becomes 0. A matrix is singular if it has an eigenvalue of 0. Therefore, if \(\lambda = 1\) is an eigenvalue of A, the matrix B is singular.

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Key Concepts

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

Matrix Singularity
A matrix is considered singular if it cannot be inverted. In other words, a singular matrix does not have a matrix inverse. This occurs when at least one of the eigenvalues of the matrix is zero.
The concept of matrix singularity is crucial in linear algebra because it affects the solutions to systems of linear equations.
  • If a matrix is singular, it might lead to systems having either no solution or infinitely many solutions.
  • Non-singular or invertible matrices, on the other hand, have unique solutions.

In our exercise, we found that if \(\lambda = 1\) is an eigenvalue of matrix \(A\), the corresponding matrix \(B\) has an eigenvalue \(\mu = 0\), indicating that \(B\) is singular. Recognizing singular matrices is key in various applications such as computer graphics and numerical simulations, where matrix inversion is common.
Eigenvalue Relationship
The relationship between eigenvalues of different matrices is a captivating topic. When a vector \(\mathbf{x}\) is an eigenvector of matrix \(A\) with eigenvalue \(\lambda\), and a new matrix \(B = I - 2A + A^2\) is defined, \(\mathbf{x}\) also becomes an eigenvector of \(B\) with a different eigenvalue \(\mu\). The relationship between the eigenvalues \(\lambda\) of \(A\) and \(\mu\) of \(B\) can be expressed as:
\[\mu = 1 - 2\lambda + \lambda^2\]
This quadratic relationship shows how the eigenvalue of matrix \(B\) depends on that of \(A\).
  • When \(\lambda\) approaches certain critical values, the behavior of \(\mu\) drastically changes, affecting the properties of matrix \(B\).
  • This relationship highlights the interconnected nature of eigenvectors and eigenvalues across transformations.

Exploring these transformations can give insights into stability and control in various scientific and engineering problems, where systems are modeled through matrices.
Matrix Algebra
Matrix algebra is a cornerstone of linear algebra and is essential for manipulating and understanding matrices. It involves operations such as addition, subtraction, scalar multiplication, and, most importantly, matrix multiplication.
Matrix multiplication is non-commutative, meaning \(AB eq BA\) in general, which requires careful attention when dealing with expressions like \(B = I - 2A + A^2\).
  • By using the distributive property and the associative property of matrix algebra, complex expressions can be simplified for analyses.
  • The calculation of powers of matrices, like \(A^2\), also plays a significant role in many transformations and model representations.

In our exercise, matrix algebra was used to derive the expression for the eigenvalues of matrix \(B\) based on those of \(A\). This showcases how fundamental matrix manipulation techniques are in deriving and proving theoretical concepts.

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Most popular questions from this chapter

Prove each of the following: (a) If \(U\) is a unit upper triangular matrix, then \(U\) is nonsingular and \(U^{-1}\) is also unit upper triangular. (b) If \(U_{1}\) and \(U_{2}\) are both unit upper triangular matrices, then the product \(U_{1} U_{2}\) is also a unit upper triangular matrix.

Let \(A\) be an \(n \times n\) matrix and let \(\lambda\) be an eigenvalue of \(A\) whose eigenspace has dimension \(k,\) where \(1

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.

For each of the following functions, determine whether the given stationary point corresponds to a local minimum, local maximum, or saddle point: (a) \(f(x, y)=3 x^{2}-x y+y^{2} \quad(0,0)\) (b) \(f(x, y)=\sin x+y^{3}+3 x y+2 x-3 y \quad(0,-1)\) (c) \(f(x, y)=\frac{1}{3} x^{3}-\frac{1}{3} y^{3}+3 x y+2 x-2 y \quad(1,-1)\) (d) \(f(x, y)=\frac{y}{x^{2}}+\frac{x}{y^{2}}+x y \quad(1,1)\) (e) \(f(x, y, z)=x^{3}+x y z+y^{2}-3 x \quad(1,0,0)\) (f) \(f(x, y, z)=-\frac{1}{4}\left(x^{-4}+y^{-4}+z^{-4}\right)+y z-x-2 y-\) \(2 z \quad(1,1,1)\)

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 \\]
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