×
×

# Solutions for Chapter 11.2: Positive Definite Matrices

## Full solutions for Linear Algebra with Applications | 1st Edition

ISBN: 9780716786672

Solutions for Chapter 11.2: Positive Definite Matrices

Solutions for Chapter 11.2
4 5 0 265 Reviews
17
2
##### ISBN: 9780716786672

Since 37 problems in chapter 11.2: Positive Definite Matrices have been answered, more than 14613 students have viewed full step-by-step solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.2: Positive Definite Matrices includes 37 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Basis for V.

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

• 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

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. 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)ยท(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 .

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• 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.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

×