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Matrices With Applications In Statistics 2nd Edition Franklin A Graybill - Solutions

10. Consider the P-matrix A below.A =$$\begin{bmatrix}3 & -2 \\3 & 4\end{bmatrix}$$.
9. Prove Theorem 11.2.2.
8. If the permutation matrix R is given by$$\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$$, show that R'AR is a P-matrix, where A is given in Prob. 6. This illustrates (4)of Theorem 11.2.2.
7. If D is given below, show that DA, AD, and A + D are P-matrices, where A is given in Prob. 6. This illustrates (i) and (5) of Theorem 11.2.2.D =$$\begin{bmatrix}1 & 0 & 0 \\0 & 3 & 0 \\0 & 0 & 5\end{bmatrix}$$
6. Show that the matrix A below is a P-matrix.A =$$\begin{bmatrix}3 & 10 & -5 \\1 & 4 & -2 \\-1 & 0 & 7\end{bmatrix}$$
5. Find upper and lower triangular matrices R and S, respectively, such that A =RS (where A is defined in Prob. 4) and such that rii > 0, sii > 0. This illustrates Theorem 11.2.1.
4. Show that the matrix A has positive leading principal minors, where A =$$\begin{bmatrix}2 & 4 & -6 \\1 & 8 & 0 \\1 & 4 & 1\end{bmatrix}$$.
3. For the matrix B, find p(B) and note that $p(B) \le p(A)$ since $B = A$, where A is defined in Prob. 1 and B is given below.$$B = \begin{bmatrix}1 & 3 & 1 \\1 & 1 & 3 \\3 & 1 & 1\end{bmatrix}$$This illustrates (4) of Theorem 11.1.1.
2. Find a characteristic vector x of the matrix A in Prob. 1 corresponding to the root $p(A)$ such that x ≥ 0.
1. Find the spectral radius $p(A)$ of the matrix A, where$$A = \begin{bmatrix}1 & 3 & 2 \\2 & 1 & 3 \\3 & 2 & 1\end{bmatrix}$$and illustrate Theorem 11.1.1.
54. Let A be a symmetric idempotent matrix. Prove that the sum of squares of the off-diagonal element of any row (or column) is less than or equal to 1/4.
53. Let A be a symmetric idempotent matrix. Prove that$$-\frac{1}{2} \le a_{ij} \le \frac{1}{2}$$ for all i ≠ j.
51. Let A and V be n x n matrices and let V be nonsingular. If AV is idempotent--- OCR End ---
50. Exhibit a 3 x 3 matrix A of rank 2 that has two roots equal to +1 and one root equal to zero such that A is not idempotent. This illustrates the fact that (1) of Theorem 12.3.2 is not a sufficient condition for A to be idempotent.
49. Prove Theorem 12.3.8.
48. Prove Theorem 12.3.7.
47. Show that Theorem 12.3.4 is not necessarily true if B is an idempotent matrix, but not symmetric.
46. In Prob. 45 let B = C−1 and partition B as$$B = \begin{bmatrix} B_{11} & b_{12} \\ b_{21} & b \end{bmatrix}$$where B₁₁ is an n x n submatrix. Show that(1) b = 0,(2) b12 = (1/n)1,(3) 1'B11 = 0,(4) AB11 and B11A are idempotent.
45. Show that C is nonsingular where C is defined by$$C = \begin{bmatrix} A & 1 \\ 1' & 0 \end{bmatrix}$$and A is defined in Prob. 43.
44. Show that A' + J is nonsingular where A is defined in Prob. 43 and k is any positive integer.
43. Let A be a symmetric n x n matrix of rank n − 1 such that 1'A = 0; that is, every column of A adds to zero. Show that B = A + (1/n)11' is nonsingular and the inverse is A' + (1/n)J.
42. If A is an n x n symmetric idempotent matrix, show that$$∑_{i} ∑_{j} a_{ij}^{2}$$ = rank (A).
41. Let A be an n x n symmetric idempotent matrix; show that, for every constant β ≠ −1, 1 + βA is nonsingular and find its inverse. If β > −1 show that 1 + βA is positive definite.
40. If A2 = aA, where a is a scalar, IX '" 0, find the scalar fJ such that IJAisidempotent.
39. Find an orthogonal matrix P such that P'(I/n)JP = D, where D is diagonal.
38. Let 1 be the n x 1 unity vector and I the k x k identity matrix. Show that A is a
37. Show that aa'/Σai2 is symmetric idempotent where a = [aij] is an n x 1 nonzero vector.
36. Let P =$$\begin{bmatrix}P1 \\P2\end{bmatrix}$$, where P is orthogonal. Show that P1P1' is idempotent.
35. If A = A' and An = An+2m-1, where m and n are any positive integers, show that A is symmetric idempotent.
34. Show that the diagonal elements of a symmetric idempotent matrix B satisfy bii ≤ 1.
33. Find a positive scalar t such that A + tB is positive definite where B is defined in Prob. 32 and A is defined below. See (3) of Theorem 12.2.14.A =$$\begin{bmatrix}3 & 1 & 0 \\1 & 1 & -1 \\0 & -1 & 4\end{bmatrix}$$
32. For the matrix B below, find a scalar t such that B + tl is positive definite. See (1) of Theorem 12.2.14.B =$$\begin{bmatrix}-3 & 2 & 0 \\2 & 1 & -1 \\0 & 1 & -2\end{bmatrix}$$
31. Prove Theorem 12.2.8.
30. Generalize the results of Prob. 29 to k x k skew cross-symmetric matrices.
29. In Prob. 28, show that AB is a C-matrix (see Def. 8.15.3), and that A + B, A - B, and A' are skew cross-symmetric matrices.
28. Akxk matrix A = [a] is defined to be skew cross-symmetric if aij = -aj+1-i, ai+1-j for all i = 1, ..., k, j = 1, 2, ..., k. Show that the matrices A and B below are skew cross-symmetric.A =$$\begin{bmatrix}0 & 1 & -3 \\0 & 0 & 0 \\3 & -1 & 0\end{bmatrix}$$; B =$$\begin{bmatrix}2 & 0 & -2 \\1 &
27. If A - B is non-negative, show that it is not always true that A2 - 82 is nonnegative.
26. If A and B are non-negative k x k matrices, show that it is not always true that x' ASx 2: O.
25. Let A be a k x k nonsingular matrix, let a be a k x 1 vector, and let α be a scalar such that α > a'A-1a ≥ 0. Show that the matrix B is nonsingular when B is defined by
24. Let A be a k x k positive definite matrix, a be a k x 1 vector, and α be a scalar such that α > a'A-1a; show that A* is a positive definite matrix when A* is defined by A* =[ A a ][ a' α ]
23. If A, B, and AB are symmetric k x k matrices and AB is nonsingular, show that there exists an orthogonal matrix P such that P'ABP, P'A-1BP, P'AB-1P and P'A-1B-1P are diagonal matrices.
22. For the matrices A and B in Prob. 19, show that(1) the characteristic roots of AB are real,(2) the roots of |A - AB| = 0 are real.This illustrates Theorem 12.2.11.
21. Show that Theorem 12.2.10 may not be true if A is not positive definite.
20. Show with an example that the converse of Theorem 12.2.9 is not true.
19. Let C be defined by *c**ij* = *a**ij**b**ij*, *i, j* = 1, 2, 3, where A and B are positive definite matrices defined below. Show that C is positive definite, and hence illustrates Theorem 12.2.8.A =[ 3 1 -1 ][ -1 2 2 ][ -1 2 4 ]B =[ 1 -1 1 ][ -1 4 1 ][ 1 1 6 ]
18. If A and I + A are k x k nonsingular matrices, show that(A + I)-1 + (A-1 + I)-1 = I.
17. In Prob. 16, show that the inverse of 2(I - A)-1 is 2(I + A)-1 - I.
16. If A is any square matrix such that I + A and I - A are nonsingular, show that?
15. If A and B are k x k nonsingular matrices that commute, show that A-1 and B-1 also commute.
14. If P is an orthogonal matrix such that P'AP is a diagonal matrix where A is a symmetric nonsingular matrix, show that P'A'P is also a diagonal matrix.
13. For any square matrix A, show that A + I and A - I commute.
12. If A is a k x k skew-symmetric matrix and A + I is nonsingular, show that B is an orthogonal matrix, where B = 2(I + A)-1 - I
11. If A is an orthogonal matrix and A + I is nonsingular, show that B is a skew-symmetric matrix, where B = 2(A + I)-1 - I
10. If A is a symmetric orthogonal matrix and A + I is nonsingular, show that A = I.
9. If A is an orthogonal k x k matrix and A + I is a nonsingular matrix, show that(A + I)-1 + [(A + I)-1]' = I.
8. Let A and A + I be nonsingular k x k matrices. Show that A-1 + I is nonsingular.
7. Find two non-negative, disjoint matrices A and B such that C = A - B where C is defined below.$$C = \begin{bmatrix}2 & 3 \\3 & 2\end{bmatrix}$$.
6. Show that the matrix A below is positive definite, and find a matrix P such that P'P = A.$$A = \begin{bmatrix}1 & 0 & -1 \\0 & 2 & 1 \\-1 & 1 & 2\end{bmatrix}$$.
5. Repeat Prob. 4, using 2(a) and 2(b) of Theorem 12.2.2.
4. For the matrices below, determine whether each is positive definite, positive semidefinite, or neither. Use 1(a) and 1(b) of Theorem 12.2.2.$$A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 4 & -2 \\ -1 & -2 & 8 \end{bmatrix}; \qquad B = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & 5 \end{bmatrix};
3. If A is a positive definite k x k matrix, show that $a_{it}a_{ts} > a_{is}^2$ for all t≠s=1, 2, ..., k.
2. Let A and B be positive definite 2 x 2 matrices. Show$a_{11}b_{11} - 2a_{12}b_{12} + a_{22}b_{22} > 0$.
1: Show that if A is a positive definite 2 x 2 matrix,(1) $a_{11} + a_{22} - 2a_{12} > 0$,(2) $a_{11} + a_{22} + 2a_{12} > 0$,(3) $a_{22}(a_{11} + a_{22} - 2a_{12}) > (a_{12} - a_{22})^2$.
46. If A is an n × n matrix, n ≥ 2, then A is an M-matrix if and only if any of the following hold.(1) (A + D)-1 > 0 for each diagonal nonsingular matrix D.(2) (A + δI)-1 > 0 for each scalar δ ≥ 0.(3) Each principal matrix of order 1, 2, and n has a positive inverse.
45. If A is a Z-matrix and all row sums are positive, then det(A) > 0.
44. If A can be written as A = δI - B, where B ≥ 0 and δ > ρ(B), then A is an M-matrix.
43. All square matrices A with aij ≤ 0 for all i ≠ j such that Ak = 0 for some positive integer k are M0-matrices.
42. If$$A = \begin{bmatrix}A_{11} & A_{12}\\0 & A_{22}\end{bmatrix}$$is a Z-matrix with positive diagonal elements, then A is an M-matrix if and only if A11 and A22 are M-matrices.
41. If A is an M-matrix, then A* is an M-matrix if and only if A* is a Z-matrix with positive diagonal elements (k is any positive integer).
40. Let A be a Z-matrix of size n X n with II S 3 and with positive diagonals. Then A is an M-matrix if and only if det(A) > O.
39. In Prob. 35, show that (I+A) > may not be true if A is reducible or if k 0.
38. Consider A in Prob. 28. Show that A-1 > 0, and thus demonstrate the result in Prob. 37.
37. If A in Prob. 35 is such that aij > 0 for all i = 1, 2, ..., n show that A-1> 0.
36. Demonstrate Prob. 35 by showing that (I + A-1)2 > 0, where A is given in Prob.28.
35. If A ≥ 0 is an n x n irreducible matrix, show that (I + A-1)-1 > 0.
34. If Ax = 0 for all x ≥ 0, show that A ≥ 0.
33. If A ≥ 0 and x ≥ y, show that Ax ≥ Ay.
32. Exhibit a vector x > 0 such that Ax = 0, where A is given in Prob. 28. This illustrates (2) of Theorem 11.5.6.
31. Show that the rank of A in Prob. 28 is 2. This illustrates (1) of Theorem 11.5.6.
30. Show that A in Prob. 28 is irreducible.
29. In Prob. 28, let ε = 0.001 and show that A + εI is an M-matrix. This illustrates(2) of Theorem 11.5.1.
28. Show that the matrix A given below is an M0-matrix but not an M-matrix.$$A =\begin{bmatrix}6 & -1 & -3 \\-2 & 1 & 0 \\-2 & -1 & 3\end{bmatrix}$$
27. Use the matrix in Prob. 22 to illustrate Corollary 11.4.8.3.
26. In Example 11.4.2, find the spectral radius of δI - A, where δ = max aij.
25. Find the spectral radius of B, where B = D-1(DA - A), where A is as defined in Prob. 23, and where$$DA =\begin{bmatrix}3 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & 4\end{bmatrix}D =\begin{bmatrix}3 & 0 & 0 \\0 & 5 & 0 \\0 & 0 & 4\end{bmatrix}$$This illustrates (2) of Theorem 11.4.6.
24. In Prob. 23, find x.
23. Show that there exists a unique vector x ~ 0 such that Ax g, where?
23. Show that there exists a unique vector x ≥ 0 such that Ax = g, where
22. Use (3) of Theorem 11.4.3 to determine if A is an M-matrix, where$$A = \begin{bmatrix} 6 & -1 & -2 \\ 0 & 2 & -1 \\ -1 & -1 & 1 \end{bmatrix}$$
21. Prove that (3) of Theorem 11.4.5 is necessary and sufficient for A to be an M-matrix.
20. Prove Theorem 11.4.2.
19. Prove (4) of Theorem 11.3.3.
18. In Prob. 15, show det(B) ≥ det(A) > 0.
17. In Prob. 15, show that each real characteristic root of B is positive.
16. Show that A⁻¹ and B⁻¹ exist, find them, and show 0 ≤ B⁻¹ ≤ A⁻¹, where A and B are as defined in Prob. 15.
15. Probs. 15, 16, 17, and 18 will illustrate Theorem 11.3.3. Define the Z-matrices A and B by (note A ≤ B)$$A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & 0 \\ -2 & -1 & 1 \end{bmatrix}; B = \begin{bmatrix} 3 & -1 & 0 \\ 0 & 2 & 0 \\ -1 & -1 & 1 \end{bmatrix}$$Show that each real characteristic of A
14. In Prob. 13, demonstrate that all off-diagonal elements of R⁻¹ and S⁻¹ are non-negative and that the diagonal elements are positive. This illustrates (4) of Theo-rem 11.3.2.
13. In Prob. 12, show that R⁻¹ and S⁻¹ exist, and find them. This illustrates (3) of Theorem 11.3.2.
12. For the Z-matrix A$$A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & 0 \\ -2 & -1 & 1 \end{bmatrix}$$find lower and upper triangular matrices R and S, respectively, each with positive diagonal elements such that A = RS.
11. Prove Theorem 11.3.1.

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