- Chapter 1: First-Order Differential Equations
- Chapter 10: Systems of Linear Differential Equations
- Chapter 11: Vector Differential Calculus
- Chapter 12: Vector Integral Calculus
- Chapter 13: Fourier Series
- Chapter 14: Fourier Series
- Chapter 15: Special Functions and Eigenfunction Expansions
- Chapter 16: Wave Motion on an Interval
- Chapter 17: The Heat Equation
- Chapter 18: The Potential Equation
- Chapter 19: Complex Numbers and Functions
- Chapter 2: Linear Second-Order Equations
- Chapter 20: Complex Integration
- Chapter 21: Complex Integration
- Chapter 22: The Residue Theorem
- Chapter 23: Conformal Mappings and Applications
- Chapter 3: The Laplace Transform
- Chapter 4: Series Solutions
- Chapter 5: Approximation of Solutions
- Chapter 6: Vectors and Vector Spaces
- Chapter 7: Matrices and Linear Systems
- Chapter 8: Determinants
- Chapter 9: Eigenvalues, Diagonalization, and Special Matrices
Advanced Engineering Mathematics 7th Edition - Solutions by Chapter
Full solutions for Advanced Engineering Mathematics | 7th Edition
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Outer product uv T
= column times row = rank one matrix.
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).
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
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