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# Solutions for Chapter 6.1: Graphs and Their Representations

## Full solutions for Mathematical Structures for Computer Science | 7th Edition

ISBN: 9781429215107

Solutions for Chapter 6.1: Graphs and Their Representations

Solutions for Chapter 6.1
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##### ISBN: 9781429215107

This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Chapter 6.1: Graphs and Their Representations includes 86 full step-by-step solutions. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. Since 86 problems in chapter 6.1: Graphs and Their Representations have been answered, more than 19980 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

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

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Column space C (A) =

space of all combinations of the columns of A.

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• 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.

• 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex 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.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• 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.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• 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.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Special solutions to As = O.

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

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

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