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# Solutions for Chapter 11.5: The Rayleigh-Ritz Method

## Full solutions for Numerical Analysis | 9th Edition

ISBN: 9780538733519

Solutions for Chapter 11.5: The Rayleigh-Ritz Method

Solutions for Chapter 11.5
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##### ISBN: 9780538733519

Chapter 11.5: The Rayleigh-Ritz Method includes 13 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Since 13 problems in chapter 11.5: The Rayleigh-Ritz Method have been answered, more than 12555 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

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.

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• 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 .

• 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.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

• Multiplier eij.

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Orthogonal subspaces.

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

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Skew-symmetric matrix K.

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

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• 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.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

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