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Chapter 6: Problem 8

Let \(A\) be a nonnegative irreducible \(3 \times 3\) matrix whose eigenvalues satisfy \(\lambda_{1}=2=\left|\lambda_{2}\right|=\left|\lambda_{3}\right|\) Determine \(\lambda_{2}\) and \(\lambda_{3}\)

Short Answer

Expert verified
In conclusion, for a nonnegative irreducible \(3 \times 3\) matrix A with eigenvalues \(\lambda_1 = 2\) and \(|\lambda_2| = |\lambda_3| = 2\), the eigenvalues are \(\lambda_2 = -2\) and \(\lambda_3 = -2\).

Step by step solution

01

Understand the properties of eigenvalues

Given that the matrix A is nonnegative and irreducible, it means that its eigenvalues will be real and nonnegative. Since 位鈧� = 2 and |位鈧倈 = |位鈧億 = 2, this implies that either 位鈧� and 位鈧� are positive or negative.
02

Use the Perron-Frobenius Theorem

According to the Perron-Frobenius Theorem for irreducible nonnegative matrices, the largest eigenvalue has a corresponding eigenvector with all positive entries. We already know that 位鈧� is the largest eigenvalue, so 位鈧� and 位鈧� must be smaller in absolute value.
03

Calculate the spectral radius

The spectral radius of a matrix is the maximum of the absolute values of its eigenvalues, and it can be denoted as 蟻(A). In our case, since 位鈧� is the largest eigenvalue and has an absolute value of 2, we have: \[蟻(A) = \max\{|位鈧亅, |位鈧倈, |位鈧億\} = 2\]
04

Analyze the possible values of 位鈧� and 位鈧�

We know that 位鈧� and 位鈧� are real and have an absolute value of 2. This means that there are only two possibilities for each: 位鈧� = 2 or 位鈧� = -2, and 位鈧� = 2 or 位鈧� = -2. However, 位鈧� = 2, so 位鈧� and 位鈧� must be distinct from 位鈧�. Therefore, we conclude that 位鈧� = -2 and 位鈧� = -2. In conclusion, the eigenvalues 位鈧� and 位鈧� of the nonnegative irreducible matrix A are both -2.

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Most popular questions from this chapter

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 \(24 \text { of Chapter } 7, \text { Section } 4 .)\) 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 a nondefective \(n \times n\) matrix with diagonalizing matrix \(X .\) Show that the matrix \(Y=\left(X^{-1}\right)^{T}\) diagonalizes \(A^{T}\)

Show that \(e^{A}\) is nonsingular for any diagonalizable matrix \(A\)

Let \(A\) be a \(n \times n\) matrix with real entries and let \(\lambda_{1}=a+b i\) (where \(a\) and \(b\) are real and \(b \neq 0\) ) be an eigenvalue of \(A .\) Let \(\mathbf{z}_{1}=\mathbf{x}+i \mathbf{y}\) (where \(\mathbf{x}\) and \(\mathbf{y}\) both have real entries) be an eigenvector belonging to \(\lambda_{1}\) and let \(\mathbf{z}_{2}=\mathbf{x}-i \mathbf{y}\) (a) Explain why \(\mathbf{z}_{1}\) and \(\mathbf{z}_{2}\) must be linearly independent. (b) Show that \(\mathbf{y} \neq \mathbf{0}\) and that \(\mathbf{x}\) and \(\mathbf{y}\) are linearly independent.

Show that if \(A\) is symmetric positive definite, then \(\operatorname{det}(A)>0 .\) Give an example of a \(2 \times 2\) matrix with positive determinant that is not positive definite.
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