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# Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation

## Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition

ISBN: 9780470383346

Solutions for Chapter 3.3: Complex Roots of the Characteristic Equation

Solutions for Chapter 3.3
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##### ISBN: 9780470383346

Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Chapter 3.3: Complex Roots of the Characteristic Equation includes 46 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 3.3: Complex Roots of the Characteristic Equation have been answered, more than 12652 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9.

Key Math Terms and definitions covered in this textbook
• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Covariance matrix:E.

When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• 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.

• 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.

• Full column rank r = n.

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

• 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.

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Pivot.

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

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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