 7.3.1: In Exercises 12, express the quadratic form in the matrix notation ...
 7.3.2: In Exercises 12, express the quadratic form in the matrix notation ...
 7.3.3: In Exercises 34, find a formula for the quadratic form that does no...
 7.3.4: In Exercises 34, find a formula for the quadratic form that does no...
 7.3.5: In Exercises 58, find an orthogonal change of variables that elimin...
 7.3.6: In Exercises 58, find an orthogonal change of variables that elimin...
 7.3.7: In Exercises 58, find an orthogonal change of variables that elimin...
 7.3.8: In Exercises 58, find an orthogonal change of variables that elimin...
 7.3.9: In Exercises 910, express the quadratic equation in the matrix form...
 7.3.10: In Exercises 910, express the quadratic equation in the matrix form...
 7.3.11: In Exercises 1112, identify the conic section represented by the eq...
 7.3.12: In Exercises 1112, identify the conic section represented by the eq...
 7.3.13: In Exercises 1316, identify the conic section represented by the eq...
 7.3.14: In Exercises 1316, identify the conic section represented by the eq...
 7.3.15: In Exercises 1316, identify the conic section represented by the eq...
 7.3.16: In Exercises 1316, identify the conic section represented by the eq...
 7.3.17: In Exercises 1718, determine by inspection whether the matrix is po...
 7.3.18: In Exercises 1718, determine by inspection whether the matrix is po...
 7.3.19: In Exercises 1924, classify the quadratic form as positive defi ni...
 7.3.20: In Exercises 1924, classify the quadratic form as positive defi ni...
 7.3.21: In Exercises 1924, classify the quadratic form as positive defi ni...
 7.3.22: In Exercises 1924, classify the quadratic form as positive defi ni...
 7.3.23: In Exercises 1924, classify the quadratic form as positive defi ni...
 7.3.24: In Exercises 1924, classify the quadratic form as positive defi ni...
 7.3.25: In Exercises 2526, show that the matrix A is positive definite firs...
 7.3.26: In Exercises 2526, show that the matrix A is positive definite firs...
 7.3.27: In Exercises 2728, use Theorem 7.3.4 to classify the matrix as posi...
 7.3.28: In Exercises 2728, use Theorem 7.3.4 to classify the matrix as posi...
 7.3.29: In Exercises 2930, find all values of k for which the quadratic for...
 7.3.30: In Exercises 2930, find all values of k for which the quadratic for...
 7.3.31: Let xT Ax be a quadratic form in the variables x1, x2,...,xn, and d...
 7.3.32: Express the quadratic form (c1x1 + c2x2 ++ cnxn)2 in the matrix not...
 7.3.33: In statistics, the quantities x = 1 n (x1 + x2 ++ xn) and s2 x = 1 ...
 7.3.34: The graph in an xyzcoordinate system of an equation of form ax2 + ...
 7.3.35: What property must a symmetric 2 2 matrix A have for xT Ax = 1 to r...
 7.3.36: Prove: If b = 0, then the cross product term can be eliminated from...
 7.3.37: Prove: If A is an n n symmetric matrix all of whose eigenvalues are...
 7.3.T1: TF. In parts (a)(l) determine whether the statement is true or fals...
 7.3.T2: TF. In parts (a)(l) determine whether the statement is true or fals...
Solutions for Chapter 7.3: Quadratic Forms
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 7.3: Quadratic Forms
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Since 39 problems in chapter 7.3: Quadratic Forms have been answered, more than 17275 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. Chapter 7.3: Quadratic Forms includes 39 full stepbystep solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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!).

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

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.

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

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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