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Chapter 8: Q14E (page 413)

The determinant of a negative definite4脳4matrix must be positive.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

The Eigen values of a negative definite matrix are negative.

Hence any 4脳4negative definite matrices will have 4 negatives (may not be distinct) Eigenvalues.

It is know that the determinant is the product of the Eigenvalues of the matrix and the product of 4 negative numbers is positive, so, the determinant of a negative definite 4脳4matrix is positive.

02

Final Answer

The given statement is TRUE.

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Most popular questions from this chapter

If A is an n脳nmatrix, what is the product of its singular values 蟽1,...,蟽n? State the product in terms of the determinant of A. For a 2脳2matrix A, explain this result in terms of the image of the unit circle.

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.

Find the singular values of A=[1224]. Find a unit vectorv鈫�such that||Av1鈫�||=蟽1. Sketch 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.

18.9x12路4x1x2+6x22=1

Cholesky factorization for 2脳2 matrices. Show that any positive definite 2脳2 matrix A can be written uniquely A=LLT as where L is a lower triangular2脳2 matrix with positive entries on the diagonal. Hint: Solve the equation

[abbc]=[x0yz][xy0z]
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