 2.1.1: If A = 1 232 7 6 5 1 0 2 3 4 , determine (a) a31, a24, a14, a32, a2...
 2.1.2: If B = 7 1 1 103 5 1 4 068 191 , determine (a) b12, b33, b41, b43, ...
 2.1.3: For 39, write the matrix with the given elements. In each case, spe...
 2.1.4: For 39, write the matrix with the given elements. In each case, spe...
 2.1.5: For 39, write the matrix with the given elements. In each case, spe...
 2.1.6: For 39, write the matrix with the given elements. In each case, spe...
 2.1.7: For 39, write the matrix with the given elements. In each case, spe...
 2.1.8: For 39, write the matrix with the given elements. In each case, spe...
 2.1.9: For 39, write the matrix with the given elements. In each case, spe...
 2.1.10: For 1012, determine tr(A) for the given matrix.A = 1 02 3 .
 2.1.11: For 1012, determine tr(A) for the given matrix.
 2.1.12: For 1012, determine tr(A) for the given matrix.A =20 132 50 1 5.
 2.1.13: For 1315, write the column vectors and row vectors of the given mat...
 2.1.14: For 1315, write the column vectors and row vectors of the given mat...
 2.1.15: For 1315, write the column vectors and row vectors of the given mat...
 2.1.16: If a1 = [1 2], a2 = [3 4], and a3 = [5 1], write the matrix A = a1 ...
 2.1.17: If a1 = [204 1 1] and a2 = [9 4 408], write the matrix A = a1 a2 , ...
 2.1.18: If b1 = 2 6 3 1 2 and b2 = 4 6 0 0 1 , write the matrix B = [b1, b2...
 2.1.19: If b1 = 2 1 4 , b2 = 5 7 6 , b3 = 0 0 0 , b4 = 1 2 3 , write the ma...
 2.1.20: If a1, a2,..., ap are each column qvectors, what are the dimension...
 2.1.21: For 2126, give an example of a matrix of the specified form. (In so...
 2.1.22: For 2126, give an example of a matrix of the specified form. (In so...
 2.1.23: For 2126, give an example of a matrix of the specified form. (In so...
 2.1.24: For 2126, give an example of a matrix of the specified form. (In so...
 2.1.25: For 2126, give an example of a matrix of the specified form. (In so...
 2.1.26: For 2126, give an example of a matrix of the specified form. (In so...
 2.1.27: For 2730, given an example of a matrix function of the specified fo...
 2.1.28: For 2730, given an example of a matrix function of the specified fo...
 2.1.29: For 2730, given an example of a matrix function of the specified fo...
 2.1.30: For 2730, given an example of a matrix function of the specified fo...
 2.1.31: Construct distinct matrix functions A and B defined on all of R suc...
 2.1.32: Show that an n n symmetric upper triangular matrix is diagonal. [Hi...
 2.1.33: Show that if A is an nn matrix that is both symmetric and skewsymm...
Solutions for Chapter 2.1: Matrices: Definitions and Notation
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 2.1: Matrices: Definitions and Notation
Get Full SolutionsSince 33 problems in chapter 2.1: Matrices: Definitions and Notation have been answered, more than 20086 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Matrices: Definitions and Notation includes 33 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations, edition: 4.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

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

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.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).