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Chapter 8: Q16E (page 413)

Matrix [-2111-2111-2]is negative definite.

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

For a matrix to be a negative definite, all the Eigenvalues of the matrix must be less than 0.

The quadratic form can be expanded as

q(x1,x2,x3)=-2x12-2x22-2x33+2x1x2+2x1x3+2x2x3

Re-write the above equation as follows:

q(x1,x2,x3)=-(x1-x2)2-(x1-x3)2-(x2-x3)2

Observe thatq1,1,1=0 which means that q is not a negative definite.

Since 0 is not less than zero the matrix cannot be negative definite. Thus, the given matrix is a non-negative definite.

02

Final Answer

The given statement is FALSE.

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Most popular questions from this chapter

Consider a singular value decomposition A=U危VTof an n脳mmatrix Awith rankA=m. Let v鈫�1,鈥�,v鈫�mbe the columns of Vandu鈫�1,鈥�,u鈫�n the columns of U. Without using the results of Chapter 5 , compute (ATA)-1ATu鈫�i. Explain the result in terms of leastsquares approximations.

Consider an matrix A, an orthogonaln脳nmatrix S, and an orthogonal m脳mmatrix R. Compare the singular values of A with those of SAR.

All positive definite matrices are invertible.

Consider the linear transformationT(q(x1x2))=q(x1,2x2) fromQ2toQ2 . Find all the eigenvalues and eigenfunctions of . Is transformation diagonalizable?

Diagonalize the13脳13matrix

[000鈥�01000鈥�01000鈥�01鈰�鈰�鈰�鈰�鈰�鈰�000鈥�01111鈰�11]

(All ones in the last row and the last column, and zeros elsewhere.)
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