8.2E: Orthogonal Diagonalization Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercises for 1
solutions
2
Normalize the rows to make each of the following matrices orthogonal.
A=[11−11] A=[3−443] A=[12−42] A=[ab−ba], (a,b)≠(0,0) A=[cosθ−sinθ0sinθcosθ0002] A=[21−11−11011] A=[−1222−1222−1] A=[26−3326−632]
- 15[3−443]
- 1√a2+b2[ab−ba]
- [2√61√6−1√61√3−1√31√301√21√2]
- 17[26−3326−632]
If P is a triangular orthogonal matrix, show that P is diagonal and that all diagonal entries are 1 or −1.
We have PT=P−1; 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=P−1, 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 P−1AP is diagonal.
A=[0110] A=[1−1−11] A=[300022025] A=[307050703] A=[110110002] A=[5−2−4−28−2−4−25] A=[5300350000710017] A=[35−11531−1−11351−153]
- 1√2[1−111]
- 1√2[011√20001−1]
- 13√2[2√231√20−42√2−31] or 13[2−2112221−2]
- 12[1−1√20−11√20−1−10√2110√2]
Consider A=[0a0a0c0c0] where one of a,c≠0. Show that cA(x)=x(x−k)(x+k), where k=√a2+c2 and find an orthogonal matrix P such that P−1AP is diagonal.
P=1√2k[c√2aa0k−k−a√2cc]
Consider A=[00a0b0a00]. Show that cA(x)=(x−b)(x−a)(x+a) and find an orthogonal matrix P such that P−1AP is diagonal.
Given A=[baab], show that
cA(x)=(x−a−b)(x+a−b) and find an orthogonal matrix P such that P−1AP is diagonal.
Consider A=[b0a0b0a0b]. Show that cA(x)=(x−b)(x−b−a)(x−b+a) and find an orthogonal matrix P such that P−1AP is diagonal.
In each case find new variables y1 and y2 that diagonalize the quadratic form q.
q=x21+6x1x2+x22 q=x21+4x1x2−2x22
- y1=1√5(−x1+2x2) and y2=1√5(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].]
- ⇒ a. By Theorem [thm:024227] let P−1AP=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=(PDP−1)2=PD2P−1=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 A∘∼B) if B=PTAP for an orthogonal matrix P.
- Show that A∘∼A for all A; A∘∼B⇒B∘∼A; and A∘∼B and B∘∼C⇒A∘∼C.
- Show that the following are equivalent for two symmetric matrices A and B.
- A and B are similar.
- A and B are orthogonally similar.
- A and B have the same eigenvalues.
Assume that A and B are orthogonally similar (Exercise [ex:8_2_12]).
- If A and B are invertible, show that A−1 and B−1 are orthogonally similar.
- Show that A2 and B2 are orthogonally similar.
- Show that, if A is symmetric, so is B.
- If B=PTAP=P−1, 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 k≥1).
If A has real eigenvalues, show that A=B+C where B is symmetric and C is nilpotent.
Let P be an orthogonal matrix.
- Show that detP=1 or detP=−1.
- Give 2×2 examples of P such that detP=1 and detP=−1.
- If detP=−1, show that I+P has no inverse. [Hint: PT(I+P)=(I+P)T.]
- [Hint: PT(I−P)=−(I−P)T.]
-
det[cosθ−sinθsinθcosθ]=1
and det[cosθsinθsinθ−cosθ]=−1[Remark: These are the only 2×2 examples.]
- Use the fact that P−1=PT to show that PT(I−P)=−(I−P)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].)
- If E is a projection matrix, show that P=I−2E is orthogonal and symmetric.
-
If P is orthogonal and symmetric, show that
E=12(I−P) is a projection matrix. - 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 A−1 is aji‖rj‖2.
We have AAT=D, where D is diagonal with main diagonal entries ‖R1‖2,…,‖Rn‖2. Hence A−1=ATD−1, and the result follows because D−1 has diagonal entries 1/‖R1‖2,…,1/‖Rn‖2.
- Let A be an m×n matrix. Show that the following are equivalent.
- A has orthogonal rows.
- A can be factored as A=DP, where D is invertible and diagonal and P has orthonormal rows.
- AAT is an invertible, diagonal matrix.
- 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.
- 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 x∙x=xTx, and use the fact that Ax=−x and A2x=x.]
- Show that P=(I−A)(I+A)−1 is orthogonal.
- Show that every orthogonal matrix P such that I+P is invertible arises as in part (b) from some skew-symmetric matrix A.
- Because I−A and I+A commute, PPT=(I−A)(I+A)−1[(I+A)−1]T(I−A)T=(I−A)(I+A)−1(I−A)−1(I+A)=I.
Show that the following are equivalent for an n×n matrix P.
- P is orthogonal.
- ‖Px‖=‖x‖ for all columns x in Rn.
- ‖Px−Py‖=‖x−y‖ for all columns x and y in Rn.
- [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 [cosθ−sinθsinθcosθ] or [cosθsinθsinθ−cosθ] for some angle θ.
Use Theorem [thm:024503] to show that every symmetric matrix is orthogonally diagonalizable.