 6.1.1: In Exercises 16, determine which of x1, x2, and x3 is an eigenvecto...
 6.1.2: In Exercises 16, determine which of x1, x2, and x3 is an eigenvecto...
 6.1.3: In Exercises 16, determine which of x1, x2, and x3 is an eigenvecto...
 6.1.4: In Exercises 16, determine which of x1, x2, and x3 is an eigenvecto...
 6.1.5: In Exercises 16, determine which of x1, x2, and x3 is an eigenvecto...
 6.1.6: In Exercises 16, determine which of x1, x2, and x3 is an eigenvecto...
 6.1.7: In Exercises 710, use the characteristic polynomial to determineif ...
 6.1.8: In Exercises 710, use the characteristic polynomial to determineif ...
 6.1.9: In Exercises 710, use the characteristic polynomial to determineif ...
 6.1.10: In Exercises 710, use the characteristic polynomial to determineif ...
 6.1.11: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.12: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.13: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.14: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.15: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.16: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.17: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.18: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.19: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.20: In Exercises 1120, find a basis for the eigenspace of A associatedw...
 6.1.21: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.22: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.23: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.24: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.25: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.26: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.27: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.28: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.29: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.30: In Exercises 2130, find the characteristic polynomial, the eigenval...
 6.1.31: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 6.1.32: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 6.1.33: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 6.1.34: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 6.1.35: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 6.1.36: FIND AN EXAMPLE For Exercises 3136, find an example thatmeets the g...
 6.1.37: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.38: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.39: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.40: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.41: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.42: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.43: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.44: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.45: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.46: TRUE OR FALSE For Exercises 3746, determine if the statementis true...
 6.1.47: Suppose that A is a square matrix with characteristic polynomial( 3...
 6.1.48: Suppose that A is a square matrix with characteristic polynomial( 1...
 6.1.49: Let T : Rn Rn be given by T(x) = Ax. Prove that if 0 isnot an eigen...
 6.1.50: Prove that if is an eigenvalue of A, then 4 is an eigenvalueof 4A.
 6.1.51: Prove that if = 1 is an eigenvalue of an n n matrix A, thenA In is ...
 6.1.52: Prove that if u is an eigenvector of A, then u is also an eigenvect...
 6.1.53: Prove that u cannot be an eigenvector associated with twodistinct e...
 6.1.54: Prove that if 5 is an eigenvalue of A, then 25 is an eigenvalueof A2.
 6.1.55: If u = 0 was allowed to be an eigenvector, then which valuesof woul...
 6.1.56: Suppose that A is a square matrix that is either upper or lowertria...
 6.1.57: Let A be an invertible matrix. Prove that if is an eigenvalueof A w...
 6.1.58: Let A =a bc d. Find a formula for the eigenvalues of A interms of a...
 6.1.59: Suppose that A and B are both n n matrices, and that u isan eigenve...
 6.1.60: Suppose that A is an n n matrix with eigenvalue and associatedeigen...
 6.1.61: Suppose that the entries of each row of a square matrix A addto zer...
 6.1.62: Suppose that A =a bc d, where a, b,c, and d satisfy a+b =c + d. Sho...
 6.1.63: Suppose that A is a square matrix. Prove that if is an eigenvalueof...
 6.1.64: Suppose that A is an n n matrix and c is a scalar. Prove thatif is ...
 6.1.65: Suppose that the entries of each row of a square matrix A addto c f...
 6.1.66: Let A be an n n matrix.(a) Prove that the characteristic polynomial...
 6.1.67: C In Exercises 6770, find the eigenvalues and bases for theeigenspa...
 6.1.68: C In Exercises 6770, find the eigenvalues and bases for theeigenspa...
 6.1.69: C In Exercises 6770, find the eigenvalues and bases for theeigenspa...
 6.1.70: C In Exercises 6770, find the eigenvalues and bases for theeigenspa...
Solutions for Chapter 6.1: Eigenvalues and Eigenvectors
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 6.1: Eigenvalues and Eigenvectors
Get Full SolutionsChapter 6.1: Eigenvalues and Eigenvectors includes 70 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. Since 70 problems in chapter 6.1: Eigenvalues and Eigenvectors have been answered, more than 13031 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: 1.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

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

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.

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

lAII = 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.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.

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