 11.2.1: In 18, the general solution of the linear system X AX is given.
 11.2.2: In 18, the general solution of the linear system X AX is given.
 11.2.3: In 18, the general solution of the linear system X AX is given.
 11.2.4: In 18, the general solution of the linear system X AX is given.
 11.2.5: In 18, the general solution of the linear system X AX is given.
 11.2.6: In 18, the general solution of the linear system X AX is given.
 11.2.7: In 18, the general solution of the linear system X AX is given.
 11.2.8: In 18, the general solution of the linear system X AX is given.
 11.2.9: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.10: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.11: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.12: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.13: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.14: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.15: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.16: In 916, classify the critical point (0, 0) of the given linear syst...
 11.2.17: Determine conditions on the real constant so that (0, 0) is a cente...
 11.2.18: Determine a condition on the real constant so that (0, 0) is a stab...
 11.2.19: Show that (0, 0) is always an unstable critical point of the linear...
 11.2.20: Let X X(t) be the response of the linear dynamical system x ax by y...
 11.2.21: Show that the nonhomogeneous linear system X AX F has a unique crit...
 11.2.22: In Example 4(b) show that (0, 0) is a stable node when bc 1.
 11.2.23: In 2326, a nonhomogeneous linear system X AX F is given. (a) In eac...
 11.2.24: In 2326, a nonhomogeneous linear system X AX F is given. (a) In eac...
 11.2.25: In 2326, a nonhomogeneous linear system X AX F is given. (a) In eac...
 11.2.26: In 2326, a nonhomogeneous linear system X AX F is given. (a) In eac...
Solutions for Chapter 11.2: Stability of Linear Systems
Full solutions for Advanced Engineering Mathematics  6th Edition
ISBN: 9781284105902
Solutions for Chapter 11.2: Stability of Linear Systems
Get Full SolutionsChapter 11.2: Stability of Linear Systems includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Since 26 problems in chapter 11.2: Stability of Linear Systems have been answered, more than 38841 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902.

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).

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.

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

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

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 .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·