 10.2.10.1.31: In 18 the general solution of the linear system X AX is given. (a) ...
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 10.2.10.1.38: In 18 the general solution of the linear system X AX is given. (a) ...
 10.2.10.1.39: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.40: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.41: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.42: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.43: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.44: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.45: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.46: In 916 classify the critical point (0, 0) of the given linear syste...
 10.2.10.1.47: Determine conditions on the real constant m so that (0, 0) is a cen...
 10.2.10.1.48: Determine a condition on the real constant m so that (0, 0) is a st...
 10.2.10.1.49: Show that (0, 0) is always an unstable critical point of the linear...
 10.2.10.1.50: Let X X(t) be the response of the linear dynamical system y x y x x...
 10.2.10.1.51: Show that the nonhomogeneous linear system X AX F has a unique crit...
 10.2.10.1.52: In Example 4(b) show that (0, 0) is a stable node when bc 1.
 10.2.10.1.53: In 2326 a nonhomogeneous linear system X AX F is given. (a) In each...
 10.2.10.1.54: In 2326 a nonhomogeneous linear system X AX F is given. (a) In each...
 10.2.10.1.55: In 2326 a nonhomogeneous linear system X AX F is given. (a) In each...
 10.2.10.1.56: In 2326 a nonhomogeneous linear system X AX F is given. (a) In each...
Solutions for Chapter 10.2: Plane Autonomous Systems
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 10.2: Plane Autonomous Systems
Get Full SolutionsDifferential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Since 26 problems in chapter 10.2: Plane Autonomous Systems have been answered, more than 20092 students have viewed full stepbystep solutions from this chapter. Chapter 10.2: Plane Autonomous Systems includes 26 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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

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).

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).

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)ยท(b  Ax) = o.

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

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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