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Chapter 8: Q40E (page 414)

For every symmetric n脳nmatrixA there exists a constantksuch thatA+klnis positive definite.

Short Answer

Expert verified

The given statement is 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

Check whether the given statement is TRUE or FALSE

It is given that,Ais symmetric and hence orthogonally diagonalizable.

Write A=UDUTand simplify as follows:

A+kln=UDUT+UklnUTA+kln=UD+klnUT

Here, D+klnis a diagonal matrix. If kis large enough, then all the diagonal entries of D+klnare positive. That is, all the eigenvalues of D+klnare positive.

All eigenvalues of A+klnare also positive since A+klnis similar to D+kln.

Thus, for large enough k,A+kln is positive definite.

03

Final Answer

The given statement is TRUE.

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a. Consider a complex upper triangularn脳nmatrix U with zeros on the diagonal. Show that u is nilpotent (i.e., thatlocalid="1659674833080" Un=0). Compare with Exercises 3.3.78 and 3.3.79.

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