- Chapter 11:
- Chapter 12:
- Chapter 13:
- Chapter 2:
- Chapter 4:
- Chapter 5:
- Chapter 6:
- Chapter 7:
- Chapter 8:
- Chapter 9:
Fundamentals of Differential Equations and Boundary Value Problems 6th Edition - Solutions by Chapter
Full solutions for Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition
Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition - Solutions by ChapterGet Full Solutions
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
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.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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!).
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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