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Chapter 8: Q2E (page 413)

The equation 2x2+5xy+3y2=1 defines an ellipse.

Short Answer

Expert verified

The statement is FALSE.

Step by step solution

01

The Quadratic form

The matrix of the quadratic form is

A=252523

The characteristic polynomial of matrix is:

A=2-位3-位-254=位2-5位+6-254=位2-5位-14

02

Find TRUE or FALSE

One of the roots of the polynomial 5卤262 mentioned above is positive, whereas the other one is negative.

As a result, the eigenvalues have opposing signs, indicating that a hyperbola is the level set defined by the quadratic form.

Hence, the given statement is FALSE.

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Most popular questions from this chapter

Consider an indefinite quadratic form q on R3with symmetric matrixA. If det A < 0 describe the level surface q(x鈫�)=0 .

Let Rbe a complex upper triangular nxn matrix with |rii|<fori,...,n. Show that

limt鈫�鈭�Rt=0,,

meaning that the modulus of all entries of Rtapproaches zero. Hint: We can write |R|鈮�(ln+U), for some positive real number |位|<1and an upper triangular U > 0 matrixwith zeros on the diagonal. Exercises 47 and 48 are helpful.

Find the singular values ofA=[100-2].

Consider a symmetric matrixA. If the vectorv鈬赌 is in the image of Aand w鈬赌 is in the kernel of A, isv鈬赌 necessarily orthogonal tow鈬赌? Justify your answer.

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]
See all solutions

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