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Chapter 8: Q24E (page 400)

Consider a quadratic form

q(x鈬赌)=x鈬赌.Ax鈬赌

where A is a symmetricnxnmatrix. Findq(e鈬赌1). Give your answer in terms of the entries of the matrix A.

Short Answer

Expert verified

qe鈬赌1=a11

Step by step solution

01

Given Information:

qx鈬赌=x鈬赌.Ax鈬赌

02

Determining :

Take a look at the quadratic form:

qx鈬赌=x鈬赌.Ax鈬赌

Where A is a nxn symmetric matrix Take into account this:

A=aij where1鈮�i,j鈮�n

Now, use the following formula to access qe鈬赌1:

localid="1659613659189" qe鈬赌1=e鈬赌1.Ae鈬赌1=e鈬赌1鈭�j-1naj1e鈬赌j=鈭�j-1naj1e鈬赌i.e鈬赌j鈬�qe鈬赌1=a11Sincee鈬赌1.e鈬赌j=未ij

03

Determining the Result:

qe鈬赌1=a11

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Most popular questions from this chapter

Let 位 be a real eigenvalue of an n x n matrix A. Show that

蟽n猢�|位|猢�蟽1,

where蟽1and蟽nare the largest and the smallest singular values of A, respectively.

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

6.q(x1,x2)=2x12+6x1x2+4x22

For the quadratic form q(x1x2)=8x12-4x1x2+5x22, find an orthogonal basis w鈫�1,w鈫�2of R2such that q(c1w鈫�1+c2w鈫�2)=c12+c22. Use your answer to sketch the level curve q(x~)=1. Compare with Example 4 and Figure 4 in this section. Exercise 63 is helpful.

Consider the transformationT(q(x1,x2))=q(x1,1) fromQ2toP2.IsT a linear transformation? Is it an isomorphism?

For which angles 胃can you find a basis of鈩�nsuch that the angle between any two vectors in this basis is 胃?
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