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Chapter 8: Q15E (page 392)

If A is invertible and orthogonally diagonalizable, is A-1orthogonally diagonalizable as well?

Short Answer

Expert verified

If A is invertible and orthogonally diagonalizable, then A-1is also orthogonally diagonalizable.

Step by step solution

01

Define symmetric matrix

  • In linear algebra, a symmetric matrix is a square matrix that does not change when its transpose is calculated.
  • A symmetric matrix is defined as one whose transpose is identical to the matrix itself.
  • A square matrix of size nxnis symmetric ifBT=B.
02

Find whether a matrix and its inverse are orthogonally diagonalizable

Let D=S-1ASwhere S is orthogonal and D is diagonal. This can be written as,

A=SDS-1=SDST

Now, since A is invertible, we take inverse of both sides, hence

A-1=SDST-1=ST-1D-1S-1=S鈯�-1D鈯�ST=SD鈯�ST

It should be evident by now that the inverse of a diagonal matrix is simply a diagonal matrix with reciprocals of each diagonal element. There will be no zeros in the diagonal matrix.

Thus, A-1is orthogonally diagonalizable with the same orthogonal matrix.

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Most popular questions from this chapter

For the quadratic form q(x1x2)=8x12-4x1x2+5x22, find an orthogonal basis w鈫�1,w鈫�2of R2such that q(c1w鈫�1+c2w鈫�2)=c12+c22. Use your answer to sketch the level curve q(x~)=1. Compare with Example 4 and Figure 4 in this section. Exercise 63 is helpful.

Consider the quadratic form

q(x1,x2)=ax12+bx1x2+cx22.

We define

q11=鈭�2q鈭�x12,q12=q21=鈭�2q鈭�x1鈭�x2,,q22=鈭�2q鈭�x22.

The discriminant D of q is defined as

D=det[q11q12q21q22]=q11q22-(q22)2.

The second derivative test tells us that if D androle="math" localid="1659684555469" q11 are both positive, then

q(x1,x2) has a minimum at (0, 0). Justify this fact, using the theory developed in this section.

Show that for every indefinite quadratic form q on R2, there exists an orthogonal basis w鈫�1,...,w鈫�2of such that q(c1w鈫�1+c2w鈫�2)=c12-c22, Hint: Modify the approach outlined in

Consider a singular value decomposition A=U危VTof an n脳mmatrix Awith rankA=m. Let v鈫�1,鈥�,v鈫�mbe the columns of Vandu鈫�1,鈥�,u鈫�n the columns of U. Without using the results of Chapter 5 , compute (ATA)-1ATu鈫�i. Explain the result in terms of leastsquares approximations.

If Ais an indefiniten脳n matrix, andR is a realn脳mrankn what can you say about the definiteness of the matrixRTAR?
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