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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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Most popular questions from this chapter

Let\(T:{{\rm P}_2} \to {{\rm P}_3}\) be a linear transformation that maps a polynomial \({\bf{p}}\left( t \right)\) into the polynomial \(\left( {t + 5} \right){\bf{p}}\left( t \right)\).

  1. Find the image of\({\bf{p}}\left( t \right) = 2 - t + {t^2}\).
  2. Show that \(T\) is a linear transformation.
  3. Find the matrix for \(T\) relative to the bases \(\left\{ {1,t,{t^2}} \right\}\) and \(\left\{ {1,t,{t^2},{t^3}} \right\}\).

Let\(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2},{{\bf{b}}_3}} \right\}\) be a basis for a vector space\(V\). Find \(T\left( {3{{\bf{b}}_1} - 4{{\bf{b}}_2}} \right)\) when \(T\) isa linear transformation from \(V\) to \(V\) whose matrix relative to \(B\) is

\({\left( T \right)_B} = \left( {\begin{aligned}0&{}&{ - 6}&{}&1\\0&{}&5&{}&{ - 1}\\1&{}&{ - 2}&{}&7\end{aligned}} \right)\)

Question: Let \(A = \left( {\begin{array}{*{20}{c}}{ - 6}&{28}&{21}\\4&{ - 15}&{ - 12}\\{ - 8}&a&{25}\end{array}} \right)\). For each value of \(a\) in the set \(\left\{ {32,31.9,31.8,32.1,32.2} \right\}\), compute the characteristic polynomial of \(A\) and the eigenvalues. In each case, create a graph of the characteristic polynomial \(p\left( t \right) = \det \left( {A - tI} \right)\) for \(0 \le t \le 3\). If possible, construct all graphs on one coordinate system. Describe how the graphs reveal the changes in the eigenvalues of \(a\) changes.

Let \(J\) be the \(n \times n\) matrix of all \({\bf{1}}\)’s and consider \(A = \left( {a - b} \right)I + bJ\) that is,

\(A = \left( {\begin{aligned}{*{20}{c}}a&b&b&{...}&b\\b&a&b&{...}&b\\b&b&a&{...}&b\\:&:&:&:&:\\b&b&b&{...}&a\end{aligned}} \right)\)

Use the results of Exercise \({\bf{16}}\) in the Supplementary Exercises for Chapter \({\bf{3}}\) to show that the eigenvalues of \(A\) are \(a - b\) and \(a + \left( {n - {\bf{1}}} \right)b\). What are the multiplicities of these eigenvalues?


For the matrix A,find real closed formulas for the trajectory x→(t+1)=Ax¯(t)wherex→=[01]. Draw a rough sketchA=[-0.51.5-0.61.3]
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