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# Solutions for Chapter 7.7: Complex Numbers

## Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Solutions for Chapter 7.7: Complex Numbers

Solutions for Chapter 7.7
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##### ISBN: 9780321785046

Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Since 118 problems in chapter 7.7: Complex Numbers have been answered, more than 55118 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.7: Complex Numbers includes 118 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Cyclic shift

S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• 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.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• Graph G.

Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

• 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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• 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.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Outer product uv T

= column times row = rank one matrix.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• 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.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Triangle inequality II u + v II < II u II + II v II.

For matrix norms II A + B II < II A II + II B IIĀ·

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

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