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# Solutions for Chapter 1.4: Matrix Algebra

## Full solutions for Linear Algebra with Applications | 8th Edition

ISBN: 9780136009290

Solutions for Chapter 1.4: Matrix Algebra

Solutions for Chapter 1.4
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##### ISBN: 9780136009290

Since 36 problems in chapter 1.4: Matrix Algebra have been answered, more than 6801 students have viewed full step-by-step solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.4: Matrix Algebra includes 36 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Adjacency matrix of a graph.

Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

• 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.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• 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.

• Left inverse A+.

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

• Linear combination cv + d w or L C jV j.

Vector addition and scalar multiplication.

• Normal matrix.

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

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Trace of A

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

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