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Chapter 7: Q-7.3-12E (page 395)

Let \(\lambda \) be any eigenvalue of a symmetric matrix \(A\). Justify the statement made in this section that \(m \le \lambda \le M\), where \(m\) and \(M\) are defined as in (2). [Hint: Find an \({\rm{x}}\) such that \(\lambda = {{\rm{x}}^T}A{\rm{x}}\).]

Short Answer

Expert verified

The statement\({{\rm{x}}^T}A{\rm{x}} = \lambda \)is justified when\(m \le \lambda \le M\).

Step by step solution

01

Symmetric Matrices and Quadratic Forms

When any Symmetric Matrix\(A\)is diagonalized orthogonallyas \(PD{P^{ - 1}}\)we have:

\(\begin{array}{l}{{\rm{x}}^T}A{\rm{x}} = {{\rm{y}}^T}D{\rm{y}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{\rm{as }}x = Py} \right\}\\{\rm{and}}\\\left\| {\rm{x}} \right\| = \left\| {P{\rm{y}}} \right\| = \left\| {\rm{y}} \right\|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\forall y \in \mathbb{R}} \right\}\end{array}\)

02

Find the maximum value

As per the question, we have:

The matrix\(A\)has eigenvalue\(\lambda \)with eigenvector\({\rm{x}}\), then:

\(\begin{array}{c}{{\rm{x}}^T}A{\rm{x}} = {{\rm{x}}^T}\lambda {\rm{x}}\\ = \lambda \left( {{{\rm{x}}^T}{\rm{x}}} \right)\\ = \lambda {\left\| {\rm{x}} \right\|^2}\\ = \lambda \end{array}\)

Thus,

\({{\rm{x}}^T}A{\rm{x}} = \lambda ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {\forall \lambda \in \left[ {m,M} \right]} \right\}\)

The statement\({{\rm{x}}^T}A{\rm{x}} = \lambda \)is justified when\(m \le \lambda \le M\).

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Most popular questions from this chapter

Question:Find the principal components of the data for Exercise 1.

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.

13. \(\left( {\begin{aligned}{{}}3&1\\1&{\,\,3}\end{aligned}} \right)\)

Question: Mark Each statement True or False. Justify each answer. In each part, A represents an \(n \times n\) matrix.

  1. If A is orthogonally diagonizable, then A is symmetric.
  2. If A is an orthogonal matrix, then A is symmetric.
  3. If A is an orthogonal matrix, then \(\left\| {A{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
  4. The principal axes of a quadratic from \({{\bf{x}}^T}A{\bf{x}}\) can be the columns of any matrix P that diagonalizes A.
  5. If P is an \(n \times n\) matrix with orthogonal columns, then \({P^T} = {P^{ - {\bf{1}}}}\).
  6. If every coefficient in a quadratic form is positive, then the quadratic form is positive definite.
  7. If \({{\bf{x}}^T}A{\bf{x}} > {\bf{0}}\) for some x, then the quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is positive definite.
  8. By a suitable change of variable, any quadratic form can be changed into one with no cross-product term.
  9. The largest value of a quadratic form \({{\bf{x}}^T}A{\bf{x}}\), for \(\left\| {\bf{x}} \right\| = {\bf{1}}\) is the largest entery on the diagonal A.
  10. The maximum value of a positive definite quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is the greatest eigenvalue of A.
  11. A positive definite quadratic form can be changed into a negative definite form by a suitable change of variable \({\bf{x}} = P{\bf{u}}\), for some orthogonal matrix P.
  12. An indefinite quadratic form is one whose eigenvalues are not definite.
  13. If P is an \(n \times n\) orthogonal matrix, then the change of variable \({\bf{x}} = P{\bf{u}}\) transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form whose matrix is \({P^{ - {\bf{1}}}}AP\).
  14. If U is \(m \times n\) with orthogonal columns, then \(U{U^T}{\bf{x}}\) is the orthogonal projection of x onto ColU.
  15. If B is \(m \times n\) and x is a unit vector in \({\mathbb{R}^n}\), then \(\left\| {B{\bf{x}}} \right\| \le {\sigma _{\bf{1}}}\), where \({\sigma _{\bf{1}}}\) is the first singular value of B.
  16. A singular value decomposition of an \(m \times n\) matrix B can be written as \(B = P\Sigma Q\), where P is an \(m \times n\) orthogonal matrix and \(\Sigma \) is an \(m \times n\) diagonal matrix.
  17. If A is \(n \times n\), then A and \({A^T}A\) have the same singular values.

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.

10. \({\bf{2}}x_{\bf{1}}^{\bf{2}} + {\bf{6}}{x_{\bf{1}}}{x_{\bf{2}}} - {\bf{6}}x_{\bf{2}}^{\bf{2}}\)

Question: Let \({x_1}\,,{x_2}\) denote the variables for the two-dimensional data in Exercise 1. Find a new variable \({y_1}\) of the form \({y_1} = {c_1}{x_1} + {c_2}{x_2}\), with\(c_1^2 + c_2^2 = 1\), such that \({y_1}\) has maximum possible variance over the given data. How much of the variance in the data is explained by \({y_1}\)?
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