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Chapter 8: Q8.1-40E (page 393)

For every symmetric n xnmatrix A there exists a constant ksuch that A+kInis positive definite.

Short Answer

Expert verified

TRUE

Step by step solution

01

Step 1: Definition of positive definite matrix

A matrix is said to be positive definite matrix, when it is symmetric matrix and all its eigenvalues are positive.

02

Step 2: Check whether the given statement is TRUE or FALSE

A is symmetric and hence orthogonally diagonalizable. Write A=UDUT.

A+kIn=UDUT+UkInUTA+kIn=UD+kInUT

D+kInis a diagonal matrix. If k is large enough, then all the diagonal entries of D+kInare positive. That is, all the eigenvalues of D+kInare positive.

All eigenvalues of A+kInalso are positive since A+kInis similar to D+kIn. Thus, for large enough k,A+kInis positive definite.

03

Step 3: Final Answer

TRUE

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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 the linear transformation

T=(q(x1,x2))=x1鈭�q鈭�x2+x2鈭�q鈭�x1

from Q2toQ2. Find all the eigenvalues and eigenfunctions of T . Is transformation T diagonalizable?

Show that an SVDA=U危VT

can be written asA=蟽1u鈫�1v鈫�1T+鈥�+蟽ru鈫�rv鈫�rT.

If A is an invertible 2x2matrix, what is the relationship between the singular values of A and A-1? Justify your answer in terms of the image of the unit circle.

Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.

15.6x12+4x1x2+3x22=1

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