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In Example 2, with A given in equation (36) above, it was claimed that equation (16) is
Chapter 7, Problem 16(choose chapter or problem)
In Example 2, with A given in equation (36) above, it was claimed that equation (16) is solvable even though the matrix A - 2I is singular. This problem justifies that claim.
a. Find all eigenvalues and eigenvectors for \(\mathbf{A}^{*}\), the adjoint of A.
b. Show that the eigenvectors of A and the corresponding eigenvectors of \(\mathbf{A}^{*}\) are orthogonal.
c. Explain why this proves that equation (16) is solvable.
Eigenvalues of Multiplicity 3. If the matrix A has an eigenvalue of algebraic multiplicity 3, then there may be either one, two, or three corresponding linearly independent eigenvectors. The general solution of the system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\) is different, depending on the number of eigenvectors associated with the triple eigenvalue. As noted in the text, there is no difficulty if there are three eigenvectors, since then there are three independent solutions of the form \(\mathbf{x}=\boldsymbol{\xi} e^{r t}\). The following two problems illustrate the solution procedure for a triple eigenvalue with one or two eigenvectors, respectively.
Questions & Answers
QUESTION:
In Example 2, with A given in equation (36) above, it was claimed that equation (16) is solvable even though the matrix A - 2I is singular. This problem justifies that claim.
a. Find all eigenvalues and eigenvectors for \(\mathbf{A}^{*}\), the adjoint of A.
b. Show that the eigenvectors of A and the corresponding eigenvectors of \(\mathbf{A}^{*}\) are orthogonal.
c. Explain why this proves that equation (16) is solvable.
Eigenvalues of Multiplicity 3. If the matrix A has an eigenvalue of algebraic multiplicity 3, then there may be either one, two, or three corresponding linearly independent eigenvectors. The general solution of the system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\) is different, depending on the number of eigenvectors associated with the triple eigenvalue. As noted in the text, there is no difficulty if there are three eigenvectors, since then there are three independent solutions of the form \(\mathbf{x}=\boldsymbol{\xi} e^{r t}\). The following two problems illustrate the solution procedure for a triple eigenvalue with one or two eigenvectors, respectively.
ANSWER:Step 1 of 6
Given:- 
