 7.5.1: Write the complex number z = 3  3/ in polar form.
 7.5.2: Find all complex numbers z such that z4 = 1. Represent your answers...
 7.5.3: For an arbitrary positive integer n, find all complex numbers c suc...
 7.5.4: Show that if c is a nonzero complex number, then there are exactly ...
 7.5.5: Show that if z is a nonzero complex number, then there exist exactl...
 7.5.6: If z is a nonzero complex number in polar form, describe 1/z in pol...
 7.5.7: Describe the transformation T(z) = (I geometrically.
 7.5.8: Use de Moivres formula to express cos(30) and sin(30) in terms of c...
 7.5.9: Consider the complex number z = 0.8 0.7/. Represent the powers z2, ...
 7.5.10: Prove the fundamental theorem of algebra for cubic polynomials with...
 7.5.11: Express the polynomial f(k) = X3  3k2 + Ik  5 as a product of lin...
 7.5.12: Consider a polynomial f(k) with real coefficients. Show that if a c...
 7.5.13: For the matrices A listed in Exercises 13 through 17, find aninvert...
 7.5.14: For the matrices A listed in Exercises 13 through 17, find aninvert...
 7.5.15: For the matrices A listed in Exercises 13 through 17, find aninvert...
 7.5.16: For the matrices A listed in Exercises 13 through 17, find aninvert...
 7.5.17: For the matrices A listed in Exercises 13 through 17, find aninvert...
 7.5.18: Consider a real 2 x 2 matrix A with two distinct real eigenvalues, ...
 7.5.19: Consider a subspace V of R", with dim(V) = m < n. a. If the n x n m...
 7.5.20: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.21: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.22: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.23: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.24: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.25: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.26: Find all complex eigenvalues of the matrices in Exercises 20 throug...
 7.5.27: Suppose a real 3 x 3 matrix A has only two distinct eigenvalues. Su...
 7.5.28: Suppose a 3 x 3 matrix A has the real eigenvalue 2 and two complex ...
 7.5.29: Consider a matrix of the formA =where a, /?, c, and d are positive ...
 7.5.30: A real nxn matrix A is called a regular transition matrix if all en...
 7.5.31: Form a 5 x 5 matrix by writing the integers 1, 2, 3,4, 5 into each ...
 7.5.32: Most longdistance telephone service in the United States is provid...
 7.5.33: The power method forf inding eigenvalues. Consider Exercises 30 and...
 7.5.34: Exercise 33 illustrates how you can use the powers of a matrix to f...
 7.5.35: Demonstrate the formulatr A = k j f k.2 + + kn,where the A/ are t...
 7.5.36: In 1990, the population of the African country Benin was about 4.6 ...
 7.5.37: Consider the set IHI of all complex 2 x 2 matrices of the formA =z...
 7.5.38: Consider the matrix'0 0 0 1' 1 0 0 0 0 1 0 0 0 0 i 0a. Find the pow...
 7.5.39: Consider the n x n matrix Cn which has ones directly below the main...
 7.5.40: Consider a cubic equationx3 + px = qywhere (p/3)3 + (q/2)2 is negat...
 7.5.41: In his high school final examination (Aarau, Switzerland, 1896), yo...
 7.5.42: Consider a complex n x m matrix A. The conjugate A is defined by ta...
 7.5.43: Consider two real nxn matrices A and B that are similar over C: Tha...
 7.5.44: Show that every complex 2 x 2 matrix is similar to an upper triangu...
 7.5.45: For which values of the real constant a are the matrices in Exercis...
 7.5.46: For which values of the real constant a are the matrices in Exercis...
 7.5.47: For which values of the real constant a are the matrices in Exercis...
 7.5.48: For which values of the real constant a are the matrices in Exercis...
 7.5.49: For which values of the real constant a are the matrices in Exercis...
 7.5.50: For which values of the real constant a are the matrices in Exercis...
 7.5.51: For Exercises 51 through 55 state whether the given set is a field ...
 7.5.52: For Exercises 51 through 55 state whether the given set is a field ...
 7.5.53: For Exercises 51 through 55 state whether the given set is a field ...
 7.5.54: For Exercises 51 through 55 state whether the given set is a field ...
 7.5.55: For Exercises 51 through 55 state whether the given set is a field ...
Solutions for Chapter 7.5: Complex Eigenvalues
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 7.5: Complex Eigenvalues
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 7.5: Complex Eigenvalues includes 55 full stepbystep solutions. Since 55 problems in chapter 7.5: Complex Eigenvalues have been answered, more than 14638 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

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.

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

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).