 6.5.1: 113 are about tests for positive definiteness.
 6.5.2: 113 are about tests for positive definiteness.
 6.5.3: 113 are about tests for positive definiteness.
 6.5.4: 113 are about tests for positive definiteness.
 6.5.5: 113 are about tests for positive definiteness.
 6.5.6: 113 are about tests for positive definiteness.
 6.5.7: 113 are about tests for positive definiteness.
 6.5.8: 113 are about tests for positive definiteness.
 6.5.9: 113 are about tests for positive definiteness.
 6.5.10: 113 are about tests for positive definiteness.
 6.5.11: 113 are about tests for positive definiteness.
 6.5.12: 113 are about tests for positive definiteness.
 6.5.13: 113 are about tests for positive definiteness.
 6.5.14: 1420 are about applications of the tests.
 6.5.15: 1420 are about applications of the tests.
 6.5.16: 1420 are about applications of the tests.
 6.5.17: 1420 are about applications of the tests.
 6.5.18: 1420 are about applications of the tests.
 6.5.19: 1420 are about applications of the tests.
 6.5.20: 1420 are about applications of the tests.
 6.5.21: 2124 use the eigenvalues; 2527 are based on pivots.
 6.5.22: 2124 use the eigenvalues; 2527 are based on pivots.
 6.5.23: 2124 use the eigenvalues; 2527 are based on pivots.
 6.5.24: 2124 use the eigenvalues; 2527 are based on pivots.
 6.5.25: With positive pivots in D, the factorization A L D L T becomes L,JD...
 6.5.26: In the Cholesky factorization A = eTc, with c T = L,JD, the square ...
 6.5.27: In the Cholesky factorization A = eTc, with c T = L,JD, the square ...
 6.5.28: W hi' I' A [cos e It out mu tIP ymg = . II smo  sin e ] [2 0] [ co...
 6.5.29: For F1(x,y) = lX4 + x 2y + y2 and F2(x,y) = x3 + xy  x find the s...
 6.5.30: The graph of z = x 2 + y2 is a bowl opening upward. The graph of z ...
 6.5.31: Which values of c give a bowl and which c give a saddle point for t...
 6.5.32: A group of nonsingular matrices includes A B and A 1 if it include...
 6.5.33: When A and B are symmetric positive definite, A B might not even be...
 6.5.34: Write down the 5 by 5 sine matrix S from Worked Example 6.5 D, cont...
 6.5.35: Suppose C is positive definite (so y T C Y > 0 whenever y =f. 0) an...
Solutions for Chapter 6.5: Positive Definite Matrices
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 6.5: Positive Definite Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. Chapter 6.5: Positive Definite Matrices includes 35 full stepbystep solutions. Since 35 problems in chapter 6.5: Positive Definite Matrices have been answered, more than 8145 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(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 .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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 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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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