 8.3.1: Find the singular values of A =
 8.3.2: Let A be an orthogonal 2 x 2 matrix. Use the image of the unit circ...
 8.3.3: Let A be an orthogonal nxn matrix. Find the singular values of A al...
 8.3.4: Find the singular values of A
 8.3.5: Find the singular values of A = Explain your answer geometrically.
 8.3.6: Find the singular values of A =1 2in 2 4. Find a unitvector 5i such...
 8.3.7: Find singular value decompositions for the matrices listed in Exerc...
 8.3.8: Find singular value decompositions for the matrices listed in Exerc...
 8.3.9: Find singular value decompositions for the matrices listed in Exerc...
 8.3.10: Find singular value decompositions for the matrices listed in Exerc...
 8.3.11: Find singular value decompositions for the matrices listed in Exerc...
 8.3.12: Find singular value decompositions for the matrices listed in Exerc...
 8.3.13: Find singular value decompositions for the matrices listed in Exerc...
 8.3.14: Find singular value decompositions for the matrices listed in Exerc...
 8.3.15: If A is an invertible 2x2 matrix, what is the relationship between ...
 8.3.16: II A is an invertible n x n matrix, what is the relationship betwee...
 8.3.17: Consider an n x m matrix A with rank(A) = m, and a singular value d...
 8.3.18: Use the result of Exercise 17 to find the leastsquares solution of...
 8.3.19: Consider an n x tn matrix A of rank r, and a singular value decompo...
 8.3.20: a. Explain how any square matrix A can he written asA = QS.where Q ...
 8.3.21: Find a polar decomposition A = QS as discussed in 6 2l 7 6 'S(C)),...
 8.3.22: Consider the standard matrix A representing the linear transformati...
 8.3.23: Consider an SVDA = UY.Vof an n x m matrix A. Show that the columns ...
 8.3.24: If A is a symmetric nxn matrix, what is the relationship between th...
 8.3.25: Let A be a 2 x 2 matrix and u a unit vector in R2. Show that<72 < ...
 8.3.26: Let A be an n x m matrix and 5 a vector in ! thatom II51 < Hit'll ...
 8.3.27: Let X he a real eigenvalue of an n x n matrix A. Show thatOn < W < ...
 8.3.28: If >4 is an n x n matrix, what is the product of its singular value...
 8.3.29: Show that an SVD A = UY.Vcan be written as
 8.3.30: Find a decomposition_ _ t T A=cr\U\V\ + ATOM 212for A = . (See Exer...
 8.3.31: Show that any matrix of rank r can be written as the sum of r matri...
 8.3.32: Consider an n x m matrix A. an orthogonal n x n matrix S, and an or...
 8.3.33: If the singular values of an n x n matrix A are all 1. is A necessa...
 8.3.34: For which square matrices A is there a singular value decomposition...
 8.3.35: Consider a singular value decomposition A = (J'LV7 of an n x m matr...
 8.3.36: Consider a singular value decomposition A = UYV1 of an n x m matrix...
Solutions for Chapter 8.3: Singular Values
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 8.3: Singular Values
Get Full SolutionsChapter 8.3: Singular Values includes 36 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since 36 problems in chapter 8.3: Singular Values have been answered, more than 14864 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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!).

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Lucas numbers
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.

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

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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).