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8.2E: Orthogonal Diagonalization Exercises

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Exercises for 1

solutions

2

Normalize the rows to make each of the following matrices orthogonal.

A=[1111] A=[3443] A=[1242] A=[abba], (a,b)(0,0) A=[cosθsinθ0sinθcosθ0002] A=[211111011] A=[122212221] A=[263326632]

  1. 15[3443]
  2. 1a2+b2[abba]
  3. [26161613131301212]
  4. 17[263326632]

If P is a triangular orthogonal matrix, show that P is diagonal and that all diagonal entries are 1 or 1.

We have PT=P1; this matrix is lower triangular (left side) and also upper triangular (right side–see Lemma [lem:006547]), and so is diagonal. But then P=PT=P1, so P2=I. This implies that the diagonal entries of P are all ±1.

If P is orthogonal, show that kP is orthogonal if and only if k=1 or k=1.

If the first two rows of an orthogonal matrix are (13,23,23) and (23,13,23), find all possible third rows.

For each matrix A, find an orthogonal matrix P such that P1AP is diagonal.

A=[0110] A=[1111] A=[300022025] A=[307050703] A=[110110002] A=[524282425] A=[5300350000710017] A=[3511531111351153]

  1. 12[1111]
  2. 12[011200011]
  3. 132[22312042231] or 13[221122212]
  4. 12[1120112011021102]

Consider A=[0a0a0c0c0] where one of a,c0. Show that cA(x)=x(xk)(x+k), where k=a2+c2 and find an orthogonal matrix P such that P1AP is diagonal.

P=12k[c2aa0kka2cc]

Consider A=[00a0b0a00]. Show that cA(x)=(xb)(xa)(x+a) and find an orthogonal matrix P such that P1AP is diagonal.

Given A=[baab], show that
cA(x)=(xab)(x+ab) and find an orthogonal matrix P such that P1AP is diagonal.

Consider A=[b0a0b0a0b]. Show that cA(x)=(xb)(xba)(xb+a) and find an orthogonal matrix P such that P1AP is diagonal.

In each case find new variables y1 and y2 that diagonalize the quadratic form q.

q=x21+6x1x2+x22 q=x21+4x1x22x22

  1. y1=15(x1+2x2) and y2=15(2x1+x2); q=3y21+2y22.

Show that the following are equivalent for a symmetric matrix A.

A is orthogonal. A2=I. All eigenvalues of A are ±1.

[Hint: For (b) if and only if (c), use Theorem [thm:024303].]

  1. a. By Theorem [thm:024227] let P1AP=D=\funcdiag(λ1,,λn) where the λi are the eigenvalues of A. By c. we have λi=±1 for each i, whence D2=I. But then A2=(PDP1)2=PD2P1=I. Since A is symmetric this is AAT=I, proving a.

[ex:8_2_12] We call matrices A and B orthogonally similar (and write AB) if B=PTAP for an orthogonal matrix P.

  1. Show that AA for all A; ABBA; and AB and BCAC.
  2. Show that the following are equivalent for two symmetric matrices A and B.
    1. A and B are similar.
    2. A and B are orthogonally similar.
    3. A and B have the same eigenvalues.

Assume that A and B are orthogonally similar (Exercise [ex:8_2_12]).

  1. If A and B are invertible, show that A1 and B1 are orthogonally similar.
  2. Show that A2 and B2 are orthogonally similar.
  3. Show that, if A is symmetric, so is B.
  1. If B=PTAP=P1, then B2=PTAPPTAP=PTA2P.

If A is symmetric, show that every eigenvalue of A is nonnegative if and only if A=B2 for some symmetric matrix B.

[ex:8_2_15] Prove the converse of Theorem [thm:024396]:

If (Ax)y=x(Ay) for all n-columns x and y, then A is symmetric.

If x and y are respectively columns i and j of In, then xTATy=xTAy shows that the (i,j)-entries of AT and A are equal.

Show that every eigenvalue of A is zero if and only if A is nilpotent (Ak=0 for some k1).

If A has real eigenvalues, show that A=B+C where B is symmetric and C is nilpotent.

Let P be an orthogonal matrix.

  1. Show that detP=1 or detP=1.
  2. Give 2×2 examples of P such that detP=1 and detP=1.
  3. If detP=1, show that I+P has no inverse. [Hint: PT(I+P)=(I+P)T.]
  4. [Hint: PT(IP)=(IP)T.]
  1. det[cosθsinθsinθcosθ]=1
    and det[cosθsinθsinθcosθ]=1

    [Remark: These are the only 2×2 examples.]

  2. Use the fact that P1=PT to show that PT(IP)=(IP)T. Now take determinants and use the hypothesis that detP(1)n.

We call a square matrix E a projection matrix if E2=E=ET. (See Exercise [ex:8_1_17].)

  1. If E is a projection matrix, show that P=I2E is orthogonal and symmetric.
  2. If P is orthogonal and symmetric, show that
    E=12(IP) is a projection matrix.

  3. If U is m×n and UTU=I (for example, a unit column in Rn), show that E=UUT is a projection matrix.

A matrix that we obtain from the identity matrix by writing its rows in a different order is called a permutation matrix. Show that every permutation matrix is orthogonal.

If the rows r1,,rn of the n×n matrix A=[aij] are orthogonal, show that the (i,j)-entry of A1 is ajirj2.

We have AAT=D, where D is diagonal with main diagonal entries R12,,Rn2. Hence A1=ATD1, and the result follows because D1 has diagonal entries 1/R12,,1/Rn2.

  1. Let A be an m×n matrix. Show that the following are equivalent.
    1. A has orthogonal rows.
    2. A can be factored as A=DP, where D is invertible and diagonal and P has orthonormal rows.
    3. AAT is an invertible, diagonal matrix.
  2. Show that an n×n matrix A has orthogonal rows if and only if A can be factored as A=DP, where P is orthogonal and D is diagonal and invertible.

Let A be a skew-symmetric matrix; that is, AT=A. Assume that A is an n×n matrix.

  1. Show that I+A is invertible. [Hint: By Theorem [thm:004553], it suffices to show that (I+A)x=0, x in Rn, implies x=0. Compute xx=xTx, and use the fact that Ax=x and A2x=x.]
  2. Show that P=(IA)(I+A)1 is orthogonal.
  3. Show that every orthogonal matrix P such that I+P is invertible arises as in part (b) from some skew-symmetric matrix A.
  1. Because IA and I+A commute, PPT=(IA)(I+A)1[(I+A)1]T(IA)T=(IA)(I+A)1(IA)1(I+A)=I.

Show that the following are equivalent for an n×n matrix P.

  1. P is orthogonal.
  2. Px=x for all columns x in Rn.
  3. PxPy=xy for all columns x and y in Rn.
  4. [Hints: For (c) (d), see Exercise [ex:5_3_14](a). For (d) (a), show that column i of P equals Pei, where ei is column i of the identity matrix.]

Show that every 2×2 orthogonal matrix has the form [cosθsinθsinθcosθ] or [cosθsinθsinθcosθ] for some angle θ.

Use Theorem [thm:024503] to show that every symmetric matrix is orthogonally diagonalizable.


8.2E: Orthogonal Diagonalization Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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