Matrix Mathematics. Errata and Addenda for the Second Edition. Dennis S. Bernstein February 13, 2014

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1 Matrix Mathematics Errata and Addenda for the Second Edition Dennis S. Bernstein February 13, 2014 This document contains an updated list of errata for the second edition of Matrix Mathematics. Please me if you discover additional errors, and I will include them in future updates. 1. Page 3, line 20: Replace and with and, respectively. 2. Page 13, Fact , statement i): Delete f 1 [f(a)]. 3. Page 24, Fact : Append ) to the left hand side of the inequality, and ] to the right hand side of the inequality. 4. Page 46, Fact : In the inequality below In particular, multiply the right hand side by Page 49, Proof of Fact : Replace obtuse with isosceles. 6. Page 51, Fact : Replace (y 2 + z 2 ) 2 with (y 2 z 2 ) Page 55, Fact : Replace complex with real. 8. Page 61, Fact : For r = 0, define M r by ( n i=1 xα i i ) 1/n. 9. Page 67, Fact : Delete the statement beginning with Now, and the following statement. 10. Page 68, Fact : Replace the last sentence with: Furthermore, equality holds if and only if there exists α 0 such that either [x 1 x n ] = α[y 1 y n ] or [y 1 y n ] = α[x 1 x n ] Page 75, Fact : In statements xix) and xx) change (Re z) 2 to 2(Re z) Page 76, Fact : In xxiii), change to =. 13. Page 83, Fact , statement xxviii): Replace tan 1 xy x+y 1 xy with tan 1 1 xy. 14. Page 105, Fact 2.5.6: Replace F n m with F n n. 15. Page 106, Proof of Proposition 2.5.9: Change the second to =. 1

2 16. Page 110, line -2: Change F m to F n. 17. Page 123, Fact : In statement i), require S F m. 18. Page 124, Fact : In statement i), delete f 1 [f(s)]. 19. Page 126, Fact , statement iii): Replace N(A ) R(B ) with N(B ) R(A ). 20. Page 128, Fact : For the two equalities, assume that both x and y are nonzero. 21. Page 131, Fact : In iv) replace the first = with = {0} and. 22. Page 137, Fact : In ii) change R to rank. 23. Page 138, Fact : In the definition of C replace (A T + yx T ) k y with (A T + yx T ) k 1 y. 24. Page 155, Fact : Replace 1 T 1 n AA 1 with 1 1 n A A 1 n Page 166, line 6: Replace nonsingular. Then, with nonsingular, then 26. Page 170, Fact : Replace 18R 2 a 2 + b 2 + c 2 with 18rR a 2 + b 2 + c Page 172, Fact , lines 6 and 12: Replace ac + bc with ac + bd. 28. Page 175, Fact , line 2: Replace [x y z] with det [x y z]. 29. Page 185, Section 3.3, line 2: Replace Section 11.5 with Section Page 193, Fact 3.7.7: Change Let A R n n. to Let A R n n, and assume that A is symmetric Page 193, Fact 3.7.9: In v) and vi), change if and only if to only if. 32. Page 194, Fact : Assume that A is skew-hermitian. 33. Pages 197 and 198, Fact and Fact : Change then ˆB and Ĉ are given by to then B and C are given by. 34. Page 203, Fact : In xliii) replace (x T x) 1 with (x T x) Page 215, Fact : Replace only if with only if A is semisimple and. 36. Page 227, Fact : In statement iii) replace = with. 37. Page 227, Fact : The solution to the problem is yes. 38. Page 231, Fact : Change p. 114 to p Page 236, Fact : Delete Toeplitz. 40. Page 237, Fact : In statement ii), change A to A Page 248: Replace c 2 + c 2 with c 2 + d Page 272, line -4: Change n = 1 to i = Page 276, Fact 4.8.2, line -5: Change β 3 to β n 3. 2

3 44. Page 276, Fact 4.8.2, line -2: Change Fact to Fact Page 300, Fact : Change perturbation to permutation twice. 46. Page 304, line 8: Replace L k+1 L k with lim k L k+1 L k. 47. Page 306, Fact : Change [174, p. 27] to [186, p. 27]. 48. Page 306, Fact : Change x > 0 and y > 0 to x >> 0 and y >> Page 320, proof of Proposition 5.4.6: Replace S by Ŝ, diag(b 1, 0, B 2 ) by diag( B 1, 0, B 2 ), I ν (A) by I ν (A), and I ν+ (A) by I ν+ (A). 50. Page 328, replace (5.6.9) by: {σ 1,..., σ n } ms = { λ 1 (A),..., λ n (A) } ms. 51. Page 333, Proposition 5.7.4: Change hold to are equivalent. 52. Page 337, Fact : Delete, and assume that S is nonsingular. 53. Page 338, Fact 5.9.2: Change min { a, b } to a 2 + b Page 342, Fact : Change mspec(i n ) = mspec(i n ) to mspec(în) = mspec(în). 55. Page 342, Fact : Change the (1,2) block of S to I. 56. Page 342, Fact : Change F n m to F n n. 57. Page 347, Fact : Change an orthogonal matrix to a projector. 58. Page 355, Fact : Delete the last sentence. 59. Page 355, Fact : Delete the left-hand inequalities in the first two strings. 60. Page 365, Fact : Assume B F m m and replace σ i (B) by σ 2 i (B) 61. Page 394, Fact : Can omit assume that A is nonsingular 62. Page 399, Fact 6.1.6: In xxx), change orthogonal to unitary. 63. Page 404, Fact 6.3.6: Change Let A F n m to Let n 2, let A F n n. 64. Page 411, Fact : Change (A I) = (A I) + to (A I) # = (A I) Page 412, Fact 6.4.2: Change Finally, to Finally, assume that A does not have full rank. Then,. 66. Page 419, Fact : Change A B to AB. 67. Page 423, Fact 6.5.5: Change and assume to and assume that A is Hermitian and. 68. Page 424, Fact 6.5.8: Change rank(d B A + B) to rank(c B A + B). 69. Page 431, Fact 6.6.3: Delete AB = Page 435, Fact : Change BA is nonsingular to CB is nonsingular. 71. Page 452, Fact : Change A, D to A, C and change C, B to B, D 72. Page 452, Fact : Change A (k) B (k) to A (n) B (n). 3

4 73. Page 456, Fact : Change x 1,..., x n R n to x 1,..., x n R. 74. Page 463, Proposition 8.2.7: In v) change β 0,..., β n 1 0 to ( 1) n i β i Page 466, proof of Corollary 8.3.7, line 2: Replace R by F. 76. Page 473, Lemma 8.5.1: Change λ nr to λ r ; change µ nr to µ r ; and change λ r R to µ r R. 77. Page 488, Fact 8.7.8: Delete and for S {1,..., n} and change inclusions to inequalities. 78. Page 502, Fact : Assume that A I is positive definite. 79. Page 505, Fact : Assume that S is nonsingular. 80. Page 505, Fact and Fact : Change H n n to H n. 81. Page 524, Fact : Delete tr AB. 82. Page 534, Fact : Change A + B on the left hand side to B 83. Page 526, Fact : The upper left inequality of the Problem is false. 84. Page 534, Fact : Change and assume that A is positive semidefinite to assume that A is positive semidefinite, and assume that B is skew Hermitian. Also, on the left hand side of the first inequality, replace A + B with B. 85. Page 535, Fact : Change det(a + ȷB) to det(a + ȷB). 86. Page 543, Fact : Change det A det Ã to det A det Ãn. 87. Page 547, Fact : Change the second instance of λ min to λ max. 88. Page 553, Fact : Change R n to F n and change min to max. 89. Page 561, Fact : Change σ i (A) to rσ i (A) 90. Page 563, Fact : Change λ i to λ i (A) twice. 91. Page 565, Fact : Change λ i [f(αa + (1 α)b)] αλ i [f(a)] + (1 α)λ i [f(b)]. to λ i [f(αa + (1 α)b)] λ i [αf(a) + (1 α)f(b)] Page 566, Fact : Multiply the left hand sides of the third and fourth strings by 1/2 93. Page 568, Fact : The answer to the problem is yes. 94. Page 572, Fact : Change spec(a) to spec(ab). 95. Page 578, Fact : Delete 1/2 on the right-hand side of the equation. 96. Page 578, Fact : In the definition of S, change A to A. Also, change A AA + A to A AA + A. 97. Page 585, Fact : Change 1 spec(a A 1 ) to 1 spec(a A T ). 98. Page 590, Fact : Change Hermitian to positive semidefinite. 4

5 99. Page 611, Corollary : Change A F m l to B F m l Page 617, Corollary 9.6.9: Assume that n m Page 618, Fact 9.7.4: In v), change the second + to Page 625, Fact : In the second string, change the fifth + to. In the last string, change the last + to Page 636, Fact 9.9.5: Change A B to B A B Page 642, Fact : Change λ i (A) to λ i, and change λ j (A) to λ j 105. Page 643, Fact : Change all to σp 106. Page 658, Fact : Change λ j to µ j 107. Page 659, Fact : Delete the second remark Page 663, Fact : Change the upper limit of the last summation to n 109. Page 665, Fact : Define r = min {m, n} Page 671, Fact : Replace A, B F n m by A F n m and B F m n 111. Page 702, Fact : Change 2sB to 2sB Page 731, first displayed equation: Change x [(ȷωI to x [(ȷωI Page 798, line 3: Change ( ) to ( ) 114. Page 809, 2 lines above (12.6.5): Change m nm to n nm 115. Page 813, Proposition : Change K R n m to K R m n. Also, the proof is missing a change of basis transformation Page 813: A 1 R q q and B k F q m. In line 4, replace A by A T Page 821, Line 6 of the proof of Proposition : Change Proposition to Proposition Page 823, Proof of Proposition : In the matrix at the bottom of the page, the entries below the sub-anti-diagonal are incorrect Page 824, Line 2: Multiply the 2 2 matrix with 1 s on the diagonal by Page 824: Proposition : Change A Rˆn ˆn to Â Rˆn ˆn Page 825: Corollary : Define S c = [O(A, C)] 1 O(A c, C c ) and S o = [O(A, C)] 1 O(A o, C o ) Page 840, line 3: Change G 2 + G 2 to G 1 + G Page 842, line above ( ): Change +ȷ cos ϕ 0 to ȷ cos ϕ Page 842, one line above ( ): Delete Re. 5

6 125. Page 852, line 2: Change P 4 to P Page 858, line 8: Change Z R n n to Z R 2n 2n Page 859, line 4: Change λz to λx Page 859, Section 12.18: Change the first paragraph to In this section we consider the existence of the maximal solution of ( ). Example shows that ( ) may not have a maximal solution. 6

CHARACTERIZATIONS. is pd/psd. Possible for all pd/psd matrices! Generating a pd/psd matrix: Choose any B Mn, then

CHARACTERIZATIONS. is pd/psd. Possible for all pd/psd matrices! Generating a pd/psd matrix: Choose any B Mn, then LECTURE 6: POSITIVE DEFINITE MATRICES Definition: A Hermitian matrix A Mn is positive definite (pd) if x Ax > 0 x C n,x 0 A is positive semidefinite (psd) if x Ax 0. Definition: A Mn is negative (semi)definite