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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
ISBN: 9780321747747
Fundamentals of Differential Equations and Boundary Value Problems | 6th Edition - Solutions by Chapter
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This textbook survival guide was created for the textbook: Fundamentals of Differential Equations and Boundary Value Problems, edition: 6. Since problems from 10 chapters in Fundamentals of Differential Equations and Boundary Value Problems have been answered, more than 1170 students have viewed full step-by-step answer. The full step-by-step solution to problem in Fundamentals of Differential Equations and Boundary Value Problems were answered by , our top Math solution expert on 11/14/17, 08:38PM. Fundamentals of Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780321747747. This expansive textbook survival guide covers the following chapters: 10.
Key Math Terms and definitions covered in this textbook
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Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
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Diagonalization
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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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.
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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!).
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Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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Orthogonal subspaces.
Every v in V is orthogonal to every w in W.
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Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.