 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 6621 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.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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 edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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).

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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