×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password

# Solutions for Chapter 8.6: Fast Fourier Transforms

## Full solutions for Numerical Analysis | 9th Edition

ISBN: 9780538733519

Solutions for Chapter 8.6: Fast Fourier Transforms

Solutions for Chapter 8.6
4 5 0 385 Reviews
17
5
##### ISBN: 9780538733519

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.6: Fast Fourier Transforms includes 10 full step-by-step solutions. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Since 10 problems in chapter 8.6: Fast Fourier Transforms have been answered, more than 13722 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9.

Key Math Terms and definitions covered in this textbook
• 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).

• 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.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• 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.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• 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.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Nilpotent matrix N.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Outer product uv T

= column times row = rank one matrix.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• 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).

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password