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Chapter 5: Q23E (page 267)

Question: Explain why a \({\bf{2}} \times {\bf{2}}\) matrix can have at most two distinct eigenvalues. Explain why an \(n \times n\) matrix can have at most n distinct eigenvalues.

Short Answer

Expert verified

A matrix \(2 \times 2\) matrix A were to have three distinct eigenvalues. This is impossible because the vectors all belong to a two-dimensional vector space and the set of vectors is linearly dependent.

If the vectors belong to n-dimensional vectors space, then p cannot exceed n.

Step by step solution

01

Write an explanation for the matrix to have two distinct eigenvalues

According to Theorem 2, a matrix \(2 \times 2\) matrix A was to have three distinct eigenvalues. This is impossible because the vectors all belong to a two-dimensional vector space and the set of vectors is linearly dependent.

02

Write an explanation for the matrix to have n distinct eigenvalues

If \(n \times n\) the matrix has \(p\) distinct values, there would be a linearly independent set of p eigenvectors.

Therefore, these vectors belong to n-dimensional vectors space, p cannot exceed n.

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