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Solutions for Chapter 7.3: Higher-Dimensional Partial Differential Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
peA) = det(A - AI) has peA) = zero matrix.
A = CTC = (L.J]))(L.J]))T for positive definite A.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.