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statistics for experimentert

Questions and Answers of Statistics For Experimentert

43. If A is defined by$$A = \begin{bmatrix} B & 0 \\\ 0 & C \end{bmatrix}$$, show that A' is a c-inverse of A where$$A' = \begin{bmatrix} B' & 0 \\\ 0 & C' \end{bmatrix}$$, where B' and C' are any
42. If A is nonsingular, show that a c-inverse of A is unique and A' = A-1.
41. Does there ever exist a c-inverse of A' that is equal to A if A is singular?
40. Show that there does not always exist a c-inverse of a matrix A' that equals A.
38. For the matrix$$A = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}$$
37. Show rank (A') ≥ rank (A) for any c-inverse of A. (See Theorem 6.6.8.)
36. Find a c-inverse of the matrix A in Prob. 6.
35. Let A and A be any two c-inverses of the *m x m* matrix A, and let g be any*n x 1* vector such that AA'g = g. Show that AA'g = g.
34. Let A be an *m x m* matrix and let P and Q be orthogonal matrices such that PAQ = D where D is a diagonal matrix. Show that QD'P' = A'.
33. If A is a positive semidefinite matrix, show that A' is also a positive semidefinite matrix.
32. Let A be an *m x n* matrix of rank *m* such that A = BC where B and C each has rank *m*. Show that (BC)' = C'B'.
31. Let A be an *m x n* matrix and let B be an *n x k* matrix. Define F and G by G = A - AB; F = AGG', and show that AB = FG and (FG)' = G'F'.
30. In Prob. 28, if PP' = I, show that P'A' is a c-inverse of AP,
29. In Prob. 28, if *k = n* and P is nonsingular, show that P'A' is a c-inverse of AP.
28. Let A be an *m x n* matrix and let P be any *n x k* matrix of rank *n*. Show that P'A' is a c-inverse of AP, where A' and P' are any c-inverses of A and P, respectively.
27. If A is an *m x n* matrix, B is an *m x n* matrix and AB' = 0, and B'A = 0, show that(1) A'B = 0,(2) B'A' = 0,(3) AB' = 0,(4) BA' = 0,(5) B'A = 0,(6) A'B' = 0.
25. If A is an m x m symmetric matrix such that a'A = 0, show that 'A = 0(a is an m x 1 vector).
24. Let λi (i = 1, 2, ..., r) be the nonzero characteristic roots of an m x m symmetric matrix A. Show that λi-1 (i = 1, 2, ..., r) are the nonzero characteristic roots of A-1.
23. Let A be an m x m symmetric matrix and P be an orthogonal matrix such that P'AP = D, where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that P'A P is also a
22. Prove Theorem 6.3.3.
21. If A is a given m x n matrix, find conditions on a matrix X so that the system AX = I is consistent.
20. Find a solution to the system of equations 2x1 - x2 + x3 = 8, x1 + 2x2 - x3 = -5.
19. Let A be an m x n given matrix, and let X be any n x m matrix such that A'AX = A'is satisfied, and let Y be any n x m matrix such that YAA' = A'is satisfied. Show that the g-inverse of A is given
18. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & -1 & 1 \\2 & -2 & 0 \\3 & 3 & -3\end{bmatrix}$$
17. Find the g-inverse of the matrix A where--- OCR End ---
16. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & 1 & 1 & 0 & 0 \\1 & 1 & 1 & 0 & 0 \\1 & 1 & 1 & 0 & 0 \\0 & 0 & 0 & 2 & 2 \\0 & 0 & 0 & 2 & 2\end{bmatrix}$$
15. Find the g-inverse of the symmetric matrix A where$$A = \begin{bmatrix}3 & 1 & 0 & 1 \\1 & 4 & -1 & 2 \\0 & -1 & 6 & 2 \\1 & 2 & 2 & 4\end{bmatrix}$$
14. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}6 & 1 & 2 & 4 & 9 \\-3 & 1 & 5 & 2 & 7 \\1 & 0 & 3 & -4 & 1 \\1 & 3 & 17 & 4 & 24 \\1 & -1 & -13 & 6 & -9\end{bmatrix}$$
13. Prove the following: Let A be an m x n matrix, X be an n x r matrix, C be an m x r matrix, B be an r x g matrix, and D be an n x g matrix. A necessary and sufficient condition that the two
12. Let the m x 2 (m≥2) matrix A of rank 1 be defined by A = [a, ca]where c is a scalar and a is an m x 1 nonzero vector. Find the g-inverse of A in terms of a and c.
10. Show that the system of equations given below is not consistent.6x1 + x2 - 3x3 + x4 = 0, 4x1 - x2 + x3 - 2x4 = 5,
9. Find the general solution to the system of equations in Prob. 7.
8. Find a solution to the system of equations in Prob. 7.
7. Use Theorem 6.3.1 to show that the system of equations given below is consistent.3x1 - 2x2 + x3 = 3, 3x1 + x2 + 2x3 = 5, 3x1 + 10x2 + 5x3 = 11.
6. Find the g-inverse of the 3 x 3 matrix A where$$A =\begin{bmatrix}3 & 2 & 1 \\1 & 1 & 1 \\3 & 1 & -1\end{bmatrix}$$using the methods presented in each of the Theorems 6.5.1, 6.5.2, 6.5.5, and
5. Show that the g-inverse of a general 2 x 2 symmetric matrix A of rank 1 defined by A =$$\begin{bmatrix}a_{11} & a_{12} \\a_{21} & a_{22}\end{bmatrix}$$is given by$$A^-
4. Find the g-inverse of the 5 x 2 matrix A where$$A = \begin{bmatrix} 2 & 4\\ 1 & 2 \\ 3 & 6 \\ 5 & 10 \\ 2 & 4 \end{bmatrix}$$Use Theorem 6.5.1.
3. Find the g-inverse of the 6 x 2 matrix A where$$A = \begin{bmatrix} 1 & 1\\ 3 & 3 \\ 5 & 2 \\ 2 & 1 \\ 0 & 6 \\ 1 & 5 \end{bmatrix}$$Use Theorem 6.2.16.
2. Find the g-inverse of the 2 x 2 matrix A where$$A = \begin{bmatrix} -1 & -1\\-1 & -1 \end{bmatrix}.$$Use Theorems 6.4.1 and 6.4.9.
1. Find the g-inverse of the vector a where$$\mathbf{a} = \begin{bmatrix} 1\\3\\1\\5\\2 \end{bmatrix}$$Use Theorem 6.4.8.
23. By the method in Prob. 22, find the inverse of I - A where$$A = \begin{bmatrix}0 & 1 & -1 \\0 & 0 & 2 \\0 & 0 & 0\end{bmatrix}.$$
22. In Prob. 21, show (I - A)-1 = Σi=0k-1 Ai.
21. Suppose A is an n x n matrix and Ak = 0 for a positive integer k. Show that I - A is nonsingular.
20. Prove Theorem 5.6.4.
19. Prove Theorem 5.6.2.
18. Prove Corollary 5.3.3.
17. In Prob. 16, find a matrix C such that AC = B.
16. Let A and B be defined by$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3\end{bmatrix}; B = \begin{bmatrix}0 & 2 & 1 \\0 & 4 & 2 \\0 & -2 & -1\end{bmatrix}.$$Show that the column space of
15. Prove Theorem 5.4.5.
14. Let A and B be m x n matrices of rank n where n < m. Show that A and B may not have the same column space.
13. If A and B are nonsingular n x n matrices, prove that they have the same column space.
12. Find the dimension of the column space of A where$$A = \begin{bmatrix}1 & 1 & 1 \\2 & 2 & 2 \\-1 & 1 & -3 \\1 & 2 & 0\end{bmatrix}$$
11. In Prob. 10, show that z₁ is the projection of y into S by using the results of Theorem 4.4.1.
10. Let S be spanned by x₁ and x₂, where x₁ = [1, 1, 0, -1]; x₂ = [0, 1, 1, 1]; the
9. Find the vector z and the scalar λ such that y = λx + z where$$y = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}; x = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}$$and such that x and z are
8. Find a basis for the orthogonal complement of the subspace in E₃ spanned by x where x' = [1, 1, -1].
7. Prove Theorem 5.2.5.
6. Use the results of Probs. 4 and 5 to demonstrate Theorem 5.2.5.
5. In Prob. 1, find the dimension of S₁ ⊕ S₂.
4. In Prob. 1, find the dimension of S₁ ∩ S₂.
3. If {a₁, ..., a,} spans S₁ and {b₁, ..., b,} spans S₂, prove that {a₁, ...,a, b₁, ..., b,} spans S₁ ⊕ S₂.
2. In Prob. 1, find a basis for S₁ ⊕ S₂.
1. If a₁, a₂, a₃ span S₁ and b₁, b₂, b₃ span S₂, find a basis for S₁ ∩ S₂.$$a₁ = \begin{bmatrix} 1 \\ 3 \\ 2 \\ -1 \end{bmatrix}; a₂ = \begin{bmatrix} 0 \\ -1 \\ 2 \\ 1
8. Find a solution to the system of equations in Prob. 7.
9. Find the general solution to the system of equations in Prob. 7.
10. Show that the system of equations given below is not consistent.6x1 + x2 - 3x3 + x4 = 0, 4x1 - x2 + x3 - 2x4 = 5,--- OCR End ---
11. In Prob, 10,show that the first three equations are consistent.
12. Let the m x 2 (m≥2) matrix A of rank 1 be defined by A = [a, ca]where c is a scalar and a is an m x 1 nonzero vector. Find the g-inverse of A in terms of a and c.
13. Prove the following: Let A be an m x n matrix, X be an n x r matrix, C be an m x r matrix, B be an r x g matrix, and D be an n x g matrix. A necessary and sufficient condition that the two
14. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}6 & 1 & 2 & 4 & 9\\-3 & 1 & 5 & 2 & 7\\1 & 0 & 3 & -4 & 1\\1 & 3 & 17 & 4 & 24\\1 & -1 & -13 & 6 & -9\end{bmatrix}$$
15. Find the g-inverse of the symmetric matrix A where$$A = \begin{bmatrix}3 & 1 & 0 & 1\\1 & 4 & -1 & 2\\0 & -1 & 6 & 2\\1 & 2 & 2 & 4\end{bmatrix}$$
7. Use Theorem 6.3.1 to show that the system of equations given below is consistent.3x1 - 2x2 + x3 = 3, 3x1 + x2 + 2x3 = 5, 3x1 + 10x2 + 5x3 = 11.
6. Find the g-inverse of the 3 x 3 matrix A where$$A = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & -1 \\ 3 & 1 & -1 \end{bmatrix}$$using the methods presented in each of the Theorems 6.5.1, 6.5.2, 6.5.5,
21. Suppose A is an n x n matrix and A^k = 0 for a positive integer k. Show that I - A is nonsingular.
22. In Prob. 21, show (I - A)^-1 = Σ^(k-1)_i=0 A^i.
23. By the method in Prob. 22, find the inverse of I - A where$$A = \begin{bmatrix}0 & 1 & -1 \\0 & 0 & 2 \\0 & 0 & 0\end{bmatrix}$$
1. Find the *g*-inverse of the vector *a* where 13*a* = 1 52 Use Theorem 6.4.8.
2. Find the *g*-inverse of the 2 x 2 matrix *A* where-1 -1*A* =-1 -1 Use Theorems 6.4.1 and 6.4.9.
3. Find the *g*-inverse of the 6 x 2 matrix *A* where 1 1 3 3 5 2*A* =2 1 0 6 1 5 Use Theorem 6.2.16.
4. Find the *g*-inverse of the 5 x 2 matrix *A* where
5. Show that the g-inverse of a general 2 x 2 symmetric matrix A of rank 1 defined by$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$is given by$$A = \begin{bmatrix} a_{11} &
16. Find the g-inverse of the matrix A where$$A = \begin{bmatrix}1 & 1 & 1 & 0 & 0\\1 & 1 & 1 & 0 & 0\\1 & 1 & 1 & 0 & 0\\0 & 0 & 0 & 2 & 2\\0 & 0 & 0 & 2 & 2\end{bmatrix}$$
17. Find the g-inverse of the matrix A where--- OCR End ---
28. Let A be an matrix and let P be any matrix of rank . Show that PA' is a c-inverse of AP, where A' and P' are any c-inverses of A and P, respectively.
29. In Prob. 28, if and P is nonsingular, show that P-1A' is a c-inverse of AP.
30. In Prob. 28, if PP' = I, show that P'A' is a c-inverse of AP.
31. Let A be an matrix and let B be an matrix. Define F and G by G = A'AB; F = AGG', and show that AB = FG and (FG)' = G'F'.
33. If A is a positive semidefinite matrix, show that A' is also a positive semidefinite matrix.
32. Let A be an matrix of rank such that A = BC where B and C each has rank . Show that (BC)' = C'B'.
34. Let A be an matrix and let P and Q be orthogonal matrices such that PAQ = D where D is a diagonal matrix. Show that QD'P = A'.
35. Let A1' and A2' be any two c-inverses of the matrix A, and let g be any vector such that AA1'g = g. Show that AA2'g = g.
27. If A is an matrix, B is an matrix and AB' = 0, and B'A = 0, show that(1) AB' = 0,(2) B'A = 0,(3) AB = 0.(4) BA' = 0,(5) B'A' = 0.(6) A'B = 0.
26. If A is an I'll x I'll symmetric matrix such that I'A = 0, sbow tha?
18. Find the g-inverse of the matrix A where$$A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & -2 & 0 \\ 3 & 3 & -3 \end{bmatrix}$$
19. Let A be an mxn given matrix, and let X be any nx m matrix such that A'AX = A'is satisfied, and let Y be any n x m matrix such that YAA' = A'is satisfied. Show that the g-inverse of A is given by
20. Find a solution to the system of equations 2x1-x2+x3=8, x1+2x2-x3=-5.
21. If A is a given mxn matrix, find conditions on a matrix X so that the system AX= I is consistent.
22. Prove Theorem 6.3.3.
23. Let A be an m x m symmetric matrix and P be an orthogonal matrix such that P'AP = D, where D is a diagonal matrix with the characteristic roots of A on the diagonal. Show that P'A P is also a
24. Let λi (i = 1,2,..., r) be the nonzero characteristic roots of an mx m symmetric matrix A. Show that λi (i = 1, 2, ..., r) are the nonzero characteristic roots of A'.

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