 5.5.1E: Let each matrix in Exercises 1–6 act on C2. Find the eigenvalues an...
 5.5.2E: Let each matrix in Exercises 1–6 act on C2. Find the eigenvalues an...
 5.5.3E: Let each matrix in Exercises 1–6 act on C2. Find the eigenvalues an...
 5.5.4E: Let each matrix in Exercises 1–6 act on C2. Find the eigenvalues an...
 5.5.5E: Let each matrix in Exercises 1–6 act on C2. Find the eigenvalues an...
 5.5.6E: Let each matrix in Exercises 1–6 act on C2. Find the eigenvalues an...
 5.5.7E: In Exercises 7–12, use Example 6 to list the eigenvalues of A. In e...
 5.5.8E: In Exercises 7–12, use Example 6 to list the eigenvalues of A. In e...
 5.5.9E: In Exercises 7–12, use Example 6 to list the eigenvalues of A. In e...
 5.5.10E: In Exercises 7–12, use Example 6 to list the eigenvalues of A. In e...
 5.5.11E: In Exercises 7–12, use Example 6 to list the eigenvalues of A. In e...
 5.5.12E: In Exercises 7–12, use Example 6 to list the eigenvalues of A. In e...
 5.5.13E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.14E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.15E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.16E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.17E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.18E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.19E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.20E: In Exercises 13–20, find an invertible matrix P and a matrix C of t...
 5.5.21E: In Example 2, solve the first equation in (2) for x2 in terms of x1...
 5.5.22E: Let A be a complex (or real) n × n matrix, and let x in Cn be an ei...
 5.5.23E: Chapter 7 will focus on matrices A with the property that AT = A. E...
 5.5.24E: Chapter 7 will focus on matrices A with the property that AT = A. E...
 5.5.25E: Let A be a real n × n matrix, and let x be a vector in Cn. Show that
 5.5.26E: Let A be a real 2 × 2 matrix with a complex eigenvalue and an assoc...
 5.5.27E: [M] In Exercises 27 and 28, find a factorization of the given matri...
 5.5.28E: [M] In Exercises 27 and 28, find a factorization of the given matri...
Solutions for Chapter 5.5: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 5.5
Get Full SolutionsChapter 5.5 includes 28 full stepbystep solutions. Since 28 problems in chapter 5.5 have been answered, more than 30922 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4.

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!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.)

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

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).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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