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Chapter 7: Q31E (page 395)

Let \(A = PD{P^{ - {\bf{1}}}}\), where P is orthogonal and D is diagonal, and let \(\lambda \) be an eigenvalue of A of multiplicity k. Then \(\lambda \) appears k times on the diagonal of D.Explain why the dimension of the eigenspace for \(\lambda \) is k.

Short Answer

Expert verified

The dimension of eigenspace is k.

Step by step solution

01

Check, if p linearly independent eigenvectors

The orthogonal matrix P, which diagonalizes matrix A is formed by linearly independent eigenvectors.

For all eigenvalues of A, there is an eigenvector.

So, for p distinct eigenvalues of A, there are p linearly independent eigenvectors

02

Find the dimension of the eigenspace of \(\lambda \)

If there are n linearly independent vectors for matrix P, then \(n - p\) the eigenvector of Pcorresponds to the remaining eigenvalues.

The remaining eigenvalues are equal to \(\lambda \) and have multiplicity k.

So, for \({\lambda _k}\), there are k linearly independent vectors.

Thus, the dimension of eigenspace is k.

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Most popular questions from this chapter

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix\(P\)and a diagonal matrix\(D\). To save you time, the eigenvalues in Exercises 17–22 are: (17)\( - {\bf{4}}\), 4, 7; (18)\( - {\bf{3}}\),\( - {\bf{6}}\), 9; (19)\( - {\bf{2}}\), 7; (20)\( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

18. \(\left( {\begin{aligned}{{}}1&{ - 6}&4\\{ - 6}&2&{ - 2}\\4&{ - 2}&{ - 3}\end{aligned}} \right)\)

(M) Orhtogonally diagonalize the matrices in Exercises 37-40. To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and for each eigenvalue \(\lambda \), find an orthogonal basis for \({\bf{Nul}}\left( {A - \lambda I} \right)\), as in Examples 2 and 3.

39. \(\left( {\begin{aligned}{{}}{.{\bf{31}}}&{.{\bf{58}}}&{.{\bf{08}}}&{.{\bf{44}}}\\{.{\bf{58}}}&{ - .{\bf{56}}}&{.{\bf{44}}}&{ - .{\bf{58}}}\\{.{\bf{08}}}&{.{\bf{44}}}&{.{\bf{19}}}&{ - .{\bf{08}}}\\{ - .{\bf{44}}}&{ - .{\bf{58}}}&{ - .{\bf{08}}}&{.{\bf{31}}}\end{aligned}} \right)\)

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

26.

  1. There are symmetric matrices that are not orthogonally diagonizable.
  2. b. If \(B = PD{P^T}\), where \({P^T} = {P^{ - {\bf{1}}}}\) and D is a diagonal matrix, then B is a symmetric matrix.
  3. c. An orthogonal matrix is orthogonally diagonizable.
  4. d. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Question: 2. Let \(\left\{ {{{\bf{u}}_1},{{\bf{u}}_2},....,{{\bf{u}}_n}} \right\}\) be an orthonormal basis for \({\mathbb{R}_n}\) , and let \({\lambda _1},....{\lambda _n}\) be any real scalars. Define

\(A = {\lambda _1}{{\bf{u}}_1}{\bf{u}}_1^T + ..... + {\lambda _n}{\bf{u}}_n^T\)

a. Show that A is symmetric.

b. Show that \({\lambda _1},....{\lambda _n}\) are the eigenvalues of A

Classify the quadratic forms in Exercises 9–18. Then make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct \(P\) using the methods of Section 7.1.

11. \({\bf{2}}x_{\bf{1}}^{\bf{2}} - {\bf{4}}{x_{\bf{1}}}{x_{\bf{2}}} - x_{\bf{2}}^{\bf{2}}\)
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