Matrix Algebra: Theory, Computations and Applications in Statistics

by James E. Gentle

Errata and Clarifications

• Page 16, line 18:
the expression should not be squared; that is, the exponent on the dot product of tx+y with itself should be removed.
(Thanks to Ismaďl Ahmed!)
• Page 18, line 7b:
should be "where \sum_i w_i = 1 and w_i >= 0."
(Thanks to Emre Celebi!)
• Page 19, line 3 (the second bullet):
the expression after the second "equals" mark should be "|a|(\sum c_i^2)^1/2"; that is, the a^2 within the summation should be deleted.
(Thanks to Rory Michaelis!)
• Page 19, line 5 (the third bullet):
the expressions on the right should be sums, such as "\rho(x)+\rho(y)", rather than products.
(Thanks to Hugo Gabriel Eyherabide!)
• Pages 19 through 25 in five displayed equations (two on p. 19, two on p.20 one of which has multiple lines, and one on p. 25):
all index initializations should be "i=1" (instead of "1=i" or even "1=1" in one case!). And I thought that I used "copy-and-paste" to reduce the number of errors!
(Thanks again to Rory Michaelis!)
• Page 20, line 9b: delete (or just ignore) the statement about "uniform" convergence; if the sequence of real numbers converges, it converges uniformly.
(Thanks to Andreas Eckner!)
• Page 23, line 10: should be "a<v_2, v_2>" instead of "a<v_1, v_1>".
(Thanks to S. J. Seyyedyazdi!)
• Page 25, line 17: "e_n" should be "e_d".
(Thanks to Dylan Keenan!)
• Page 27, line 2:
"for any nonzero scalar" should be "for any positive scalar".
(Thanks to Clark Fitzgerald!)
• Page 28, line 10b:
"\tilde{x}_i=x_i" should be "\tilde{x}_k=x_k".
(Thanks to Selene Makarios!)
• Page 32, line 6b: "is a vector space" should be "is a cone but not necessarily a convex cone".
(Thanks again to S. J. Seyyedyazdi!)
• Page 34, line 18:
"x" should be "\bar{x}"; that is, the residuals x_c and the fit \bar{x} are orthogonal.
(Thanks to Ognyan Palatov!)
• Page 35, line 12b: instead of "From equation (2.45)", it should be "From equation (2.47)"
(Thanks again to Andreas Eckner!)
• Page 38, line 1b (Exercise 2.12): "is a vector space" should be "is a cone but not necessarily a convex cone".
(Thanks again to S. J. Seyyedyazdi!)
• Page 47, line 1:
"elements" should be "element"
• Page 51, line 5:
"q-p+1" should be "q-p-1".
(Thanks to Ruolin Liu!)
• Page 51, equation (3.18):
in the first two terms, the elements in the (2,1) position of the 2 by 2 matrix should be "a_{23}", instead of "a_{32}".
(Thanks to András Keszei!)
• Page 53, line 7b: remove the "T" in "(ij)^T"
(Thanks again to Andreas Eckner!)
• Page 70, equation (3.64):
the equation is incorrect, and there is no simple re-expression of the quantity on the left side of the equation unless, of course, A is the identity, in which case the equation is correct if the "n" is removed from the right side of the equation.
(Thanks to Afante!)
• Page 73, equation (3.69): The "i" in the second index of B should be "j", and the numerators in all of the greatest integer functions should be decreased by 1; that is, the indexes on A should be [(i-1)/p]+1 and [(j-1)/q]+1, and the indexes on B should be i-p[(i-1)/p] and j-q[(j-1)/q].
The derived expressions at the bottom of page 73 and the top of page 74 should be changed accordingly. (Copy-and-paste got me again!)
(Thanks again to Andreas Eckner!)
• Page 75, line 21 (first sentence after equation (3.81)):
Delete the sentence "From equations..."; the sentence is incorrect.
(Thanks again to Clark Fitzgerald!)
• Page 77, line 9: should add the condition "a ≠ 0" (otherwise the statement is trivially true).
(Thanks again to Andreas Eckner!)
• Page 80, line 14b: "by equation (3.92)" should be "by equation (3.93)"
(Thanks again to Andreas Eckner!)
• Page 82, line 16b: "is equivalent to the condition" should be "implies the equaivalence of the conditions", and then the conditions in equation (3.106) should be
"[A|b]y = 0 for some y neq 0 iff Ax=0 for some x neq 0."
(Thanks again to S. J. Seyyedyazdi!)
• Page 84, line 8b:
"column row rank" should be "column rank"
(Thanks to Greg Fodor!)
• Page 85, line 4b:
"row column rank" should be "full row rank"
(Thanks again to Clark Fitzgerald!)
• Page 89, lines 14--16: these statements do not follow from the facts above; they are corollaries to equation (3.127) derived in Section 3.3.7, but more resaons need to be stated.
(Thanks again to Andreas Eckner!)
• Page 89, line 9b:
"(xA)^T C(Ax)" should be "(Ax)^T C(Ax)"
(Thanks again to Greg Fodor!)
• Page 90, line 3b:
"(yA)^T (Ay)" should be "(Ay)^T (Ay)"
(Thanks again to Greg Fodor!)
• Page 93, prior to equations (3.133) through (3.139):
add the condition "and such sums as I+A, A+B, and so on, are full rank" (that is, every matrix whose inverse is indicated is of full rank).
also on page 93, line 5 from the bottom, should say
"... these equations may hold ..." (also see correction for page 101 below)
(Thanks to John Angus!)
• Page 100, line 17:
"A^T B=0 and B^T A=0" should be "A B=0 and B^T A^T=0"
also on page 100, line 22:
"... A are orthogonal ..." should be "... A^T are orthogonal ..."
(Thanks to Eric!)
• Page 101, Section 3.6.1:
should say
"Equations (1.133) through (1.139) do not hold in general."
(Thanks again to John Angus!)
• Page 102, line 5b:
"row column rank" should be "full row rank"
(Thanks again to Clark Fitzgerald!)
• Page 104, line 16:
Insert at beginning of sentence, When n >= m''. (See fist paragraph in that subsection.)
(Thanks again to Clark Fitzgerald!)
• Page 104, line 6b:
1'' should be +-1'' (See line 13 on that page.)
(Thanks again to Clark Fitzgerald!)
• Page 105, line 8:
Should include the condition n >= m''. (This applies only to the orthogonal group.)
(Thanks again to Clark Fitzgerald!)
• Page 107, item 5:
should be "If A is diagonal or triangular with elements a_{ii}, the eigenvalues are a_{ii}, and for diagonal A the corresponding eigenvectors are e_i (the unit vectors)."
also on page 107, item 7:
"cd" should be "d"
(Thanks again to Eric!)
• Page 128, line 3:
The matrix D should be n by n, rather than "m by m"
(Thanks again to Greg Fodor!)
• Page 137, line 1 "equation (3.236)" should be "equation (2.44)".
(Thanks again to Andreas Eckner!)
• Page 137, equation (3.252) should be an inequality.
(Thanks again to Andreas Eckner!)
• Page 139, line 2 "k x n" should be "k x m".
(Thanks again to Andreas Eckner!)
• Page 147, line 10: the sentence should end after "continuous" at the end of the line; the references to negative reals and reals should be omitted.
(Thanks again to Andreas Eckner!)
• Page 154, Table 4.1:
the derivative of ax should be "aI";
(Thanks to Chetan Bhole!)
the derivative of x^Tb should be "b";
(Thanks to Josef Hebenstreit!)
the derivative of xx^T should be "x K I + I K x", where "K" denotes Kronecker product.
(Thanks to Chetan Bhole!)
the derivative of x^TAb should be "Ab";
(Thanks to Josef Hebenstreit!)
the derivative of V(x) as given assumes the mean of x, xbar, is 0.
the derivative in general is "2*(x - xbar/n)/(n - 1)";
(Thanks to Kamesh Kompella!)
• Page 155, equation (4.15):
the definition of the derivative of a matrix Y with respect to a matrix X is the common definition used by statisticians. It might be noted that an alternate and perhaps better definition is
dY/dX=d(vec(Y))/d(vec(X)).
• Page 159, line 6b:
"m X m-k" should be "m X (m-k)"
(Thanks to Chu Xinqi!)
• Page 171, Exercise 4.8: there should be a minus sign on the trace in the exponent.
(Thanks again to Andreas Eckner!)
• Page 174, line 21:
"(xQ)^T (Qy)" should be "(Qx)^T (Qy)"
(Thanks again to Greg Fodor!)
• Page 180, line 5:
"orthogonal" should be "orthonormal"
(Thanks again to Clark Fitzgerald!)
• Page 180, lines 5 and 15:
"u" should be "v"; that is, the reflection is about v
(Thanks to Eric Arora!)
• Page 184 equations (5.15) and (5.16) and the equation following (5.16):
Replace all occurrences of "(x_qq - 1)" with "(x_qq - a)", and remove the "2" in front of the square root in (5.16).
(Thanks again to Andreas Eckner!)
• Page 198 equations (5.44) and (5.16):
Replace "a_{ij}^{(k)}-a_{ij}^{(k)}a_{ij}^{(k)}a_{ij}^{(k)}" with
"a_{ij}^{(k)}-a_{mj}^{(k)}a_{ij}^{(k)}/a_{jj}^{(k)} where m≥ i''.
(Thanks again to Andreas Eckner!)
• Page 212, Section 6.3.1:
If necessary, the equations must be rearranged so that no diagonal element of the coefficient matrix is 0.
• Page 213, eq (6.16):
On the left hand side "(D+L)" should be "(D+omega L)".
• Page 225, line 13b:
The development is "below"; not "above".
(Thanks again to Andreas Eckner!)
• Page 225, line 6b:
The "ratio" of norms should be the "product" of norms.
(Thanks again to Andreas Eckner!)
• Page 226, line 4:
"page 290" should be "page 197" (although the fact is also mentioned on page 290).
(Thanks again to Andreas Eckner!)
• Page 226, line 1b:
There should be a final factor of Q'; that is, the equation should be
X^+ = [R_1^{-1} 0]Q^{\rm T}
(Thanks again to Greg Fodor!)
• Page 230, line 1b:
Should be "goes through the mean (4.2, 3)".
(Thanks again to Andreas Eckner!)
• Page 236, line 9:
should say "sufficient condition for a unique solution to exist is that d_{m}>d_{m+1}. (Recall ..."
If d_{m}=d_{m+1} a solution may or may not exist.
• Page 246, line 2b, inequality (7.11):
"x^(j+1)" and "x^(j)" should be "u^(j+1)" and "u^(j)".
(Thanks again to Greg Fodor!)
• Page 268, line 3b:
"n-k" should be "n-m".
(Thanks again to Andreas Eckner!)
• Page 274, line 9b:
Not all normal matrices have eigenvalues that are real. There are various conditions unstated; hence, this line should be deleted.
• Page 277, second bullet:
should be "A strictly diagonally dominant symmetric matrix is positive definite."
• Page 279, line 4:
Insert "is positive definite" immediately before "follows".
(Thanks again to Andreas Eckner!)
• Page 289, eq (8.48):
should be "XB = XC" instead of "X^TB=X^TC".
(Thanks to Wei Luo!)
• Page 317, line 19 (fourth bullet):
"for any i" should be "for some i".
(Thanks to Max van de Sande Bakhuyzen!)
• Page 340, line 12:
"Beta_W" should be "Beta_V".
(Thanks to Jaafar AlMutawa!)
• Page 343, eq (9.32):
the gamma should be transposed
(Thanks to Hal Pedersen!)
• Page 345, line 19, just after eq (9.35):
should be "The vectors v_i are the same..."
(Thanks to Andres Velazquez!)
• Page 352, following equation (9.49):
The statistic generally does not have a chi-squared distribution, as pointed out by Cragg and Donald (1996, JASA v. 91, 1301). (This is due to to the indeterminacy of the decomposition referred to in the paragraph following expression (9.49).) Cragg and Donald suggest an alternative test based on the LDU decomposition. Another, likely better, test based on the SVD was proposed by Kleibergen and Paap (2006, J. Econometrics v. 133, 97).
• Page 361, line 12b:
"x" and "y" should be reversed in the implication that defines the symmetric relationship.
(Thanks again to Clark Fitzgerald!)
• Page 378, line 7b: should be "2^{k-1}+i" (instead of "2^{k-1}-i"; i is negative).
(Thanks again to Andreas Eckner!)
• Page 380, line 18 and following:
The set of fixed-point numbers is a proper subset of the integers, but the set of floating-point numbers is not a proper subset of the reals because it contains some special numbers that are not real.
• Page 403, line 6: should be "x_u is [x]_c" instead of "x_c is [x]_c".
(Thanks again to Andreas Eckner!)
• Page 404, line 7b: should be "the result in R may not exist in F" (instead of "the result in F may not exist in R").
(Thanks again to Andreas Eckner!)
• Page 418, line 6b: should be "0<r<1" (instead of "0<+r<h").
(Thanks again to Andreas Eckner!)
• Page 472, lines 3 and 4 from bottom:
There should be two sentences:
Hilbert matrices have large condition numbers; for example, the $10 \times 10$ Hilbert matrix has condition number of order $10^{13}$. The Matlab function {\tt hilb(n)} generates an $n \times n$ Hilbert matrix.
(Thanks to chtoch!)
• Page 482, lines 14 through 17: the definitions of ceiling and floor are reversed; ⌈x⌉ is the smallest integer greater than or equal to x, and ⌊x⌋ is the largest integer less than or equal to x.
(Thanks again to Andreas Eckner!)
• Page 489, line4b:
"diagonal" should be "trace"
• Page 497, 6.2c:
This is just incorrect. The spectral radius of the original problem is 1.5. This means Gauss-Seidel will not converge. The obvious question, then, is what to do, and I had meant to add some exercises relating to that question (using pivoting). The solution given is for a different question -- just ignore it!