 5.2.1: In Exercises 14, show that A and B are not similar matrices.
 5.2.2: In Exercises 14, show that A and B are not similar matrices.
 5.2.3: In Exercises 14, show that A and B are not similar matrices.
 5.2.4: In Exercises 14, show that A and B are not similar matrices.
 5.2.5: Let A be a matrix with characteristic equation . What are the possi...
 5.2.6: Let (a) Find the eigenvalues of A. (b) For each eigenvalue , find t...
 5.2.7: In Exercises 711, use the method of Exercise 6 to determine whether...
 5.2.8: In Exercises 711, use the method of Exercise 6 to determine whether...
 5.2.9: In Exercises 711, use the method of Exercise 6 to determine whether...
 5.2.10: In Exercises 711, use the method of Exercise 6 to determine whether...
 5.2.11: In Exercises 711, use the method of Exercise 6 to determine whether...
 5.2.12: In Exercises 1215, find a matrix P that diagonalizes A, and compute .
 5.2.13: In Exercises 1215, find a matrix P that diagonalizes A, and compute .
 5.2.14: In Exercises 1215, find a matrix P that diagonalizes A, and compute .
 5.2.15: In Exercises 1215, find a matrix P that diagonalizes A, and compute .
 5.2.16: In Exercises 1621, find the geometric and algebraic multiplicity of...
 5.2.17: In Exercises 1621, find the geometric and algebraic multiplicity of...
 5.2.18: In Exercises 1621, find the geometric and algebraic multiplicity of...
 5.2.19: In Exercises 1621, find the geometric and algebraic multiplicity of...
 5.2.20: In Exercises 1621, find the geometric and algebraic multiplicity of...
 5.2.21: In Exercises 1621, find the geometric and algebraic multiplicity of...
 5.2.22: Use the method of Example 5 to compute , where
 5.2.23: Use the method of Example 5 to compute , where
 5.2.24: In each part, compute the stated power of
 5.2.25: Find if n is a positive integer and
 5.2.26: Let Show that (a) A is diagonalizable if . (b) A is not diagonaliza...
 5.2.27: In the case where the matrix A in Exercise 26 is diagonalizable, fi...
 5.2.28: Prove that similar matrices have the same rank
 5.2.29: Prove that similar matrices have the same nullity
 5.2.30: Prove that similar matrices have the same trace.
 5.2.31: Prove that if A is diagonalizable, then so is for every positive in...
 5.2.32: Prove that if A is a diagonalizable matrix, then the rank of A is t...
 5.2.33: Suppose that the characteristic polynomial of some matrix A is foun...
 5.2.34: This problem will lead you through a proof of the fact that the alg...
 5.2.a: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.b: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.c: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.d: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.e: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.f: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.g: In parts (a)(h) determine whether the statement is true or false, a...
 5.2.h: In parts (a)(h) determine whether the statement is true or false, a...
Solutions for Chapter 5.2: Diagonalization
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 5.2: Diagonalization
Get Full SolutionsSince 42 problems in chapter 5.2: Diagonalization have been answered, more than 14088 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Chapter 5.2: Diagonalization includes 42 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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 .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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 AI 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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.