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Chapter 7: Q10E (page 383)

There exists a real 5 脳 5 matrix without any real eigenvalues.

Short Answer

Expert verified

False, that without any real eigenvalues there exists a real 5 脳 5 matrix.

Step by step solution

01

Define eigenvalue:

Eigenvalues is a special set of scale values associated with a set of line numbers in almost matrix measurements.

02

Explanation for the real 5 × 5 matrix without any real eigenvalues:

If a complex number Z is an eigenvalue of a matrix , then it is a root of det (A-位濒)which is a polynomial, so its complex conjugate zmust also be an eigenvalue of . Thus, since a 5脳5 matrix has 5 eigenvalues (counting their algebraic multiplicities), then at least one of them must be a real one

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Most popular questions from this chapter

Consider an upper triangular n脳nmatrix Awithaij鈮�0fori=1,2,鈥�,mandaij=0fori=m+1,鈥�n. Find the algebraic multiplicity of the eigenvalueof. Without using Theorem 7.3.6, what can you say about the geometric multiplicity?

If a matrix A has k distinct eigenvalues, then (A)鈮�K

Find a basis of the linear space Vof all2脳2matrices Afor which[1-3] is an eigenvector, and thus determine the dimension of V.

TRUE OR FALSE

18. If A and B are nxn matrices, if 伪is an eigenvalue of A, and if 尾is an eigenvalue of B, then 伪尾 must be an eigenvalue of AB.

(a). If 2i is an eigenvalue of a real 2 脳 2 matrix A, findA2.

(b). Give an example of a real 2 脳 2 matrix A such that all the entries of A are nonzero and 2i is an eigenvalue of A. ComputeA2and check that your answer agrees with part (a).
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