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Chapter 8: Q 8.1-13E (page 391)

Question: Consider a symmetric 3 脳 3 matrix A with A2 = I3. Is the linear transformation

necessarily the reflection about a subspace of?

Short Answer

Expert verified

The linear transformation is a reflection about the subspace of .

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 is symmetric if BT= B.
02

Find the linear transformation

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Most popular questions from this chapter

54. If Aand B are real symmetric matrices such that,A3=B3 thenmust be equal to B.

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

Show that for every symmetricn脳n matrix A there exists a constant k such that matrixA+kIn is positive definite.

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.

Consider a quadratic form qon with symmetric matrix A, with rank A = r.Suppose that Ahas ppositive eigenvalues, if eigenvalues are counted with their multiplicities. Show that there exists an orthogonal basis w鈫�1,...w鈫�nofRnsuch that q(c1w鈫�1+....+cnw鈫�n)=cp2+....+cp2+....+cp2-cp+12-....-cr2..Hint: Modify the approach outlined in and 65.
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