- 3.3.1: The system in Exercise 1, Section 3.2
- 3.3.2: The system in Exercise 2, Section 3.2
- 3.3.3: The system in Exercise 3, Section 3.2
- 3.3.4: The system in Exercise 6, Section 3.2
- 3.3.5: The system in Exercise 7, Section 3.2
- 3.3.6: The system in Exercise 8, Section 3.2
- 3.3.7: The system in Exercise 9, Section 3.2
- 3.3.8: The system in Exercise 10, Section 3.2
- 3.3.9: The initial-value problems in Exercise 11, Section 3.2
- 3.3.10: The initial-value problems in Exercise 12, Section 3.2
- 3.3.11: The initial-value problems in Exercise 13, Section 3.2
- 3.3.12: The initial-value problems in Exercise 14, Section 3.2
- 3.3.13: The second-order equation in Exercise 21, Section 3.2
- 3.3.14: The second-order equation in Exercise 22, Section 3.2
- 3.3.15: The second-order equation in Exercise 23, Section 3.2
- 3.3.16: The second-order equation in Exercise 24, Section 3.2
- 3.3.17: dx dt dy dt = 2 1 0 1 x y 18
- 3.3.18: dx dt dy dt = 2 1 1 1 x y 19.
- 3.3.19: The slope field for the system dx dt = 2x + 1 2 y dy dt = y is show...
- 3.3.20: The slope field for the system dx dt = 2x + 6y dy dt = 2x 2y is sho...
- 3.3.21: For the harmonic oscillator with mass m = 1, spring constant k = 6,...
- 3.3.22: Consider a harmonic oscillator with mass m = 1, spring constant k =...
- 3.3.23: dY dt = 0.2 0.1 0.0 0.1 Y
- 3.3.24: dY dt = 0.1 0.2 0.0 1.0 Y 2
- 3.3.25: dY dt = 0.2 0.1 0.0 0.1 Y 2
- 3.3.26: dY dt = 0.1 0.0 0.2 0.2 Y 2
- 3.3.27: Consider the linear system dY dt = 2 1 0 2 Y. (a) Show that (0, 0) ...
Solutions for Chapter 3.3: PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUES
Full solutions for Differential Equations 00 | 4th Edition
Solutions for Chapter 3.3: PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUESGet Full Solutions
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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·
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.