Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 1-13 are about tests for positive definiteness.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 14-20 are about applications of the tests.

Introduction to Linear Algebra 4
Problems 21-24 use the eigenvalues; Problems 25-27 are based on pivots.

Introduction to Linear Algebra 4
Problems 21-24 use the eigenvalues; Problems 25-27 are based on pivots.

Introduction to Linear Algebra 4
Problems 21-24 use the eigenvalues; Problems 25-27 are based on pivots.

Introduction to Linear Algebra 4
Problems 21-24 use the eigenvalues; Problems 25-27 are based on pivots.

Introduction to Linear Algebra 4
With positive pivots in D, the factorization A L D L T becomes L,JD,JD LT. (Square roots of the pivots give D = ,JD J15.) Then C = J15 L T yields the Cholesky factorization A = eTc which is "symmetrized L U": From C = [~ ;] find A. From A = [: 2~] find C = chol(A).

Introduction to Linear Algebra 4
In the Cholesky factorization A = eTc, with c T = L,JD, the square roots of the pivots are on the diagonal of C. Find C (upper triangular) for

Introduction to Linear Algebra 4
In the Cholesky factorization A = eTc, with c T = L,JD, the square roots of the pivots are on the diagonal of C. Find C (upper triangular) for

Introduction to Linear Algebra 4
W hi' I' A [cos e It out mu tIP ymg = . II smo - sin e ] [2 0] [ cos e cos e 0 5 - sin e (b) the eigenvalues of A sin e) find cose ' (a) the determinant of A (c) the eigenvectors of A (d) a reason why A is symmetric positive definite.

Introduction to Linear Algebra 4
For F1(x,y) = -lX4 + x 2y + y2 and F2(x,y) = x3 + xy - x find the second derivative matrices HI and H 2: [ a2Fjax2 a2 Fj aXay ] Test for minimum. H = a2 2 2 is positive definite Fjayax a Fjay HI is positive definite so FI is concave up (= convex). Find the minimum point of Fl and the saddle point of

Introduction to Linear Algebra 4
The graph of z = x 2 + y2 is a bowl opening upward. The graph of z = x 2 - y2 is a saddle. The graph of z = _x2 - y2 is a bowl opening downward. What is a test on a, b, C for z = ax2 + 2bxy + cy2 to have a saddle point at (O,O)?

Introduction to Linear Algebra 4
Which values of c give a bowl and which c give a saddle point for the graph of z = 4x2 + 12xy + cy2? Describe this graph at the borderline value of c.

Introduction to Linear Algebra 4
A group of nonsingular matrices includes A B and A -1 if it includes A and B. "Products and inverses stay in the group." Which of these are groups (as in 2.7.37)? Invent a "subgroup" of two of these groups (not I by itself = the smallest group). (a) Positive definite symmetric matrices A. (b) Orthogo

Introduction to Linear Algebra 4
When A and B are symmetric positive definite, A B might not even be symmetric. But its eigenvalues are still positive. Start from ABx = AX and take dot products with Bx. Then prove A > O.

Introduction to Linear Algebra 4
Write down the 5 by 5 sine matrix S from Worked Example 6.5 D, containing the eigenvectors of K when n = 5 and h = 1/6. Multiply K times S to see the five positive eigenvalues. Their sum should equal the trace 10. Their product should be det K = 6.

Introduction to Linear Algebra 4
Suppose C is positive definite (so y T C Y > 0 whenever y =f. 0) and A has independent columns (so Ax =f. 0 whenever x =f. 0). Apply the energy test to X T ATCAx to show that ATCA is positive definite: the crucial matrix in engineering.