/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Which of the matrices that follo... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 6

Which of the matrices that follow are positive definite? Negative definite? Indefinite? (a) \(\left(\begin{array}{ll}3 & 2 \\ 2 & 2\end{array}\right)\) (b) \(\left(\begin{array}{ll}3 & 4 \\ 4 & 1\end{array}\right)\) (c) \(\left(\begin{array}{rr}3 & \sqrt{2} \\ \sqrt{2} & 4\end{array}\right)\) (d) \(\left(\begin{array}{rrr}-2 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & -2\end{array}\right)\) (e) \(\left(\begin{array}{lll}1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 2\end{array}\right)\) (f) \(\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 5 & 3 \\ 0 & 3 & 5\end{array}\right)\)

Short Answer

Expert verified
Matrix (a) is positive definite, (b) is indefinite, (c) is positive definite, (d) is negative definite, (e) is indefinite, and (f) is positive definite.

Step by step solution

01

positive definite

matrix. #Step 4: Find the principal minors for matrix (b)# Matrix (b) is: \(\begin{pmatrix}3 & 4 \\ 4 & 1\end{pmatrix}\) The principal minors are the determinants of the submatrices: 1st-order minors: \(D_1 = 3\) \(D_2 = 1\) 2nd-order minor: \(D_3 = \det\begin{pmatrix}3 & 4 \\ 4 & 1\end{pmatrix} = 3*1 - 4*4 = -13\) #Step 5: Determine the definiteness of matrix (b)# Matrix (b) has a negative 2nd-order minor. Therefore, matrix (b) is a
02

indefinite

matrix. #Step 6: Find the principal minors for matrix (c)# Matrix (c) is: \(\begin{pmatrix}3 & \sqrt{2} \\ \sqrt{2} & 4\end{pmatrix}\) The principal minors are the determinants of the submatrices: 1st-order minors: \(D_1 = 3\) \(D_2 = 4\) 2nd-order minor: \(D_3 = \det\begin{pmatrix}3 & \sqrt{2} \\ \sqrt{2} & 4\end{pmatrix} = 3*4 - 2 = 10\) #Step 7: Determine the definiteness of matrix (c)# All the principal minors of matrix (c) are positive. Therefore, matrix (c) is a
03

positive definite

matrix. #Step 8: Find the principal minors for matrix (d)# Matrix (d) is: \(\begin{pmatrix}-2 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & -2\end{pmatrix}\) The principal minors are the determinants of the submatrices: 1st-order minors: \(D_1 = -2\) \(D_2 = -1\) \(D_3 = -2\) 2nd-order minors: \(D_4 = \det\begin{pmatrix}-2 & 0 \\ 0 & -1\end{pmatrix} = 2\) \(D_5 = \det\begin{pmatrix}-2 & 1 \\ 1 & -2\end{pmatrix} = 3\) \(D_6 = \det\begin{pmatrix}-1 & 0 \\ 0 & -2\end{pmatrix} = 2\) 3rd-order minor: \(D_7 = \det\begin{pmatrix}-2 & 0 & 1 \\ 0 & -1 & 0 \\ 1 & 0 & -2\end{pmatrix} = -8\) #Step 9: Determine the definiteness of matrix (d)# Matrix (d) has negative odd-order minors and positive even-order minors. Therefore, matrix (d) is a
04

negative definite

matrix. #Step 10: Find the principal minors for matrix (e)# Matrix (e) is: \(\begin{pmatrix}1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 2\end{pmatrix}\) The principal minors are the determinants of the submatrices: 1st-order minors: \(D_1 = 1\) \(D_2 = 1\) \(D_3 = 2\) 2nd-order minors: \(D_4 = \det\begin{pmatrix}1 & 2 \\ 2 & 1\end{pmatrix} = -3\) \(D_5 = \det\begin{pmatrix}1 & 1 \\ 1 & 2\end{pmatrix} = 1\) \(D_6 = \det\begin{pmatrix}1 & 1 \\ 2 & 1\end{pmatrix} = -1\) 3rd-order minor: \(D_7 = \det\begin{pmatrix}1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 2\end{pmatrix} = -14\) #Step 11: Determine the definiteness of matrix (e)# Matrix (e) has a negative 2nd-order minor. Therefore, matrix (e) is a
05

indefinite

matrix. #Step 12: Find the principal minors for matrix (f)# Matrix (f) is: \(\begin{pmatrix}2 & 0 & 0 \\ 0 & 5 & 3 \\ 0 & 3 & 5\end{pmatrix}\) The principal minors are the determinants of the submatrices: 1st-order minors: \(D_1 = 2\) \(D_2 = 5\) \(D_3 = 5\) 2nd-order minors: \(D_4 = \det\begin{pmatrix}2 & 0 \\ 0 & 5\end{pmatrix} = 10\) \(D_5 = \det\begin{pmatrix}2 & 0 \\ 0 & 5\end{pmatrix} = 10\) 3rd-order minor: \(D_6 = \det\begin{pmatrix}2 & 0 & 0 \\ 0 & 5 & 3 \\ 0 & 3 & 5\end{pmatrix} = 20\) #Step 13: Determine the definiteness of matrix (f)# All the principal minors of matrix (f) are positive. Therefore, matrix (f) is a
06

positive definite

matrix. In conclusion, the definite types of the given matrices are: - (a) Positive definite - (b) Indefinite - (c) Positive definite - (d) Negative definite - (e) Indefinite - (f) Positive definite

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Most popular questions from this chapter

Let \(A\) be a \(n \times n\) matrix with Schur decomposition \(U T U^{H} .\) Show that if the diagonal entries of \(T\) are all distinct, then there is an upper triangular matrix \(R\) such that \(X=U R\) diagonalizes \(A\)

Let \(A\) be an \(n \times n\) matrix and let \(\lambda\) be an eigenvalue of \(A\) whose eigenspace has dimension \(k\) where \(1

Show that any \(3 \times 3\) matrix of the form \\[ \left(\begin{array}{lll} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & b \end{array}\right) \\] is defective.

Let \\[ A=\left(\begin{array}{rr} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array}\right) \\] (a) Show that \(A\) is positive definite and that \(\mathbf{x}^{T} A \mathbf{x}=\mathbf{x}^{T} B \mathbf{x}\) for all \(\mathbf{x} \in \mathbb{R}^{2}\) (b) Show that \(B\) is positive definite, but \(B^{2}\) is not positive definite.

Let \(A\) be an \(n \times n\) positive stochastic matrix with dominant eigenvalue \(\lambda_{1}=1\) and linearly independent eigenvectors \(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n},\) and let \(\mathbf{y}_{0}\) be an initial probability vector for a Markov chain \\[ \mathbf{y}_{0}, \mathbf{y}_{1}=A \mathbf{y}_{0}, \mathbf{y}_{2}=A \mathbf{y}_{1}, \dots \\] (a) Show that \(\lambda_{1}=1\) has a positive eigenvector \(\mathbf{x}_{1}\) (b) Show that \(\left\|\mathbf{y}_{j}\right\|_{1}=1, j=0,1, \ldots\) (c) Show that if \\[ \mathbf{y}_{0}=c_{1} \mathbf{x}_{1}+c_{2} \mathbf{x}_{2}+\cdots+c_{n} \mathbf{x}_{n} \\] then the component \(c_{1}\) in the direction of the positive eigenvector \(\mathbf{x}_{1}\) must be nonzero. (d) Show that the state vectors \(\mathbf{y}_{j}\) of the Markov chain converge to a steady-state vector. (e) Show that \\[ c_{1}=\frac{1}{\left\|\mathbf{x}_{1}\right\|_{1}} \\] and hence the steady-state vector is independent of the initial probability vector \(\mathbf{y}_{0}\)
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