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Chapter 7: Q43E (page 383)

If two matrices A and B have the same characteristic polynomials, then they must be similar.

Short Answer

Expert verified

False, that two matrices A and B have the same characteristic polynomials.

Step by step solution

01

Define polynomial matrix:

Polynomial matrix or polynomial matrix is a matrix whose elements are homogeneous or polynomial. A characteristic polynomial of a square matrix is a polynomial that is immutable under matrix equality and has eigenvalues as sources.

02

Explanation for two matrices A and B:

Consider two matrices for example,

A=1001;B=1101

These two matrices clearly have the same characteristic polynomial, which is fA(位)=fB位=(1-位)2However, for any invertible matrix S applies S-1AS=S-1l2S=l2鈮�B. Thus, A and B are not similar.

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Most popular questions from this chapter

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

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25: Consider a positive transition matrix

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Consider the matrixA=|1k11|where k is an arbitrary constant. For which values of k does A have two distinct real eigenvalues? When is there no real eigenvalue?

Find a basis of the linear space V of all 2脳2matrices Afor which both[11]and[12]are eigenvectors, and thus determine the dimension of.
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