 3.10.1: In each of 14 find the general solution of the given system of dif...
 3.10.2: In each of 14 find the general solution of the given system of dif...
 3.10.3: In each of 14 find the general solution of the given system of dif...
 3.10.4: In each of 14 find the general solution of the given system of dif...
 3.10.5: In each of 58, solve the given initialvalue problem
 3.10.6: In each of 58, solve the given initialvalue problem
 3.10.7: In each of 58, solve the given initialvalue problem
 3.10.8: In each of 58, solve the given initialvalue problem
 3.10.9: Let Show that
 3.10.10: Let Prove that Hint: Write A in the form and observe that
 3.10.11: Let A be the n x n matrix and let P be the n x n matrix(a) Show tha...
 3.10.12: Compute ek if
 3.10.13: (a) Show that eTIAT=~ 'eAT. (b) Given that with compute eh.
 3.10.14: Suppose thatp(A)= det(AAI) has n distinct roots A,, . . .,A,. Prov...
 3.10.15: Suppose that A*= aA. Find eA'.
 3.10.16: Let (a) Show that A(A  51) = 0. (b) Find eA
 3.10.17: Let (a) Show that A2 = 1.(b) Show thatA = ( cost sin t ) sint cost '
 3.10.18: In each of 1820 verify directly the CayleyHamilton Theorem for th...
 3.10.19: In each of 1820 verify directly the CayleyHamilton Theorem for th...
 3.10.20: In each of 1820 verify directly the CayleyHamilton Theorem for th...
Solutions for Chapter 3.10: Equal roots
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 3.10: Equal roots
Get Full SolutionsSince 20 problems in chapter 3.10: Equal roots have been answered, more than 5842 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. Chapter 3.10: Equal roots includes 20 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069.

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).

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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 .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

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.

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.

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

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.