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Chapter 8: Q21E (page 413)

If Ais an invertible symmetric matrix, thenA2must be positive definite.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

Let be ann脳n symmetric matrix. Then,A is diagonalizable and definite. Let位1,位2,鈰�,位n be its eigenvalues. We have位i鈮�0 for any since A is invertible.

Also,A2 is a symmetric matrix and the eigenvalues ofA2 are 位12,位22,鈰�,位n2. Since位i鈮�0 for any i, we have 位i2>0. Thus,A2 is a symmetric matrix having all positive eigenvalues which implies thatA2 is positive definite.

02

Final Answer

The given statement is TRUE.

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Most popular questions from this chapter

Let Abe anmatrix and v鈫�a vector in Rm.Show that

蟽m||v鈫�||鈮�||Av鈫�||鈮�蟽1||v鈫�||

where 蟽1and蟽mare the largest and the smallest singular values of A, respectively. Compare this with Exercise 25.

Let 位 be a real eigenvalue of an n x n matrix A. Show that

蟽n猢�|位|猢�蟽1,

where蟽1and蟽nare the largest and the smallest singular values of A, respectively.

Consider the nxnmatrix with all ones on the main diagonal and all q'selsewhere. For which values of q is this matrix invertible? Hint: Exercise 17is helpful.

51. IfAis a symmetric2脳2 matrix with eigenvalues 1 and 2, then the angle betweenx鈫�andAx鈫� must be less than蟺/6, for all nonzero vectorsx鈫�inR2.

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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