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Chapter 8: Q37E (page 414)

37. If Ais a positive definiten脳nmatrix andx鈫� is a nonzero vector inRn, then the angle betweenx鈫�andAx鈫�must be acute.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

Recall that if is the angle between two vectors x鈫�and y鈫�, then 0鈮�胃鈮�蟺and .

cos胃=y藱.y藱x藱y藱

If A is a positive definite, then x藱.Ax藱>0.

If the angle between them is 胃, then using the above formula, we get 肠辞蝉胃>0.

Here, 肠辞蝉胃is positive if 胃is acute and negative if 胃is obtuse. Hence, the angle between x藱and Ax藱is acute.

02

Final Answer

The given statement is TRUE.

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Most popular questions from this chapter

Consider an orthogonal matrix Rwhose first column isv鈬赌. Form the symmetric matrix A=v鈬赌v鈬赌T. Find an orthogonal matrix Sand a diagonal matrix Dsuch that S-1AS=D . Describe Sin terms ofR.

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