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Chapter 8: Q4E (page 400)

Determine the definiteness of the quadratic forms in Exercises 4 through 7.

4.q(x1,x2)=6x12+4x1x2=3x22

Short Answer

Expert verified

位1=7,位2=2

Step by step solution

01

Given Information:

q(x1,x2)=6x12+4x1x2=3x22

02

Split the term and analyse the quadratic form:

q(x1,x2)=6x12+4x1x2=3x22

Split the term4x1x2equally between the two components are

x1x2脳6x1+2x22x1+3x2

To analyse the quadratic form as

qx鈫�=x鈫�脳Ax鈫�

A=6223

Auxiliary equation,

det(A-位l2=det6223-位1001=0

det6-位223-位=0

6-位3-位-22=0

18-6位-3位+位2-4=0

位2-9位+14=0位-7位-2=0

A-位濒2=位-7位-2=0位1=7,位2=2

The two eigen values are positive then the matrix is positive definite

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Most popular questions from this chapter

LetU猢�0be a real upper triangularn脳n matrix with zeros on the diagonal. Show that

(ln+U)t猢�tn(ln+U+U2+.....+Un-1)

for all positive integers t. See Exercises 46 and 47.

We say that ann脳nmatrix A is triangulizable ifis similar to an upper triangular n脳nmatrix B.

a. Give an example of a matrix with real entries that fails to be triangulizable over R.

b. Show that anyn脳nmatrix with complex entries is triangulizable over C . Hint: Give a proof by induction analogous to the proof of Theorem 8.1.1.

Find the singular values of A=[1224]. Find a unit vectorv鈫�such that||Av1鈫�||=蟽1. Sketch the image of the unit circle.

For the matrix A=[8-2-25],writeA=B2 as discussed in Exercise 30. See Example 1.

If the singular values of an n脳nmatrix A are all 1 , A is necessarily orthogonal?
See all solutions

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