Page 445 - Matrix Analysis & Applied Linear Algebra

5.13 Orthogonal Projection 441

5.13.13. Let M and N be subspaces of a vector space V, and consider the
associated orthogonal projectors P M and P N .
(a) Prove that P M P N = 0 if and only if M⊥N.
(b) Is it true that P M P N = 0 if and only if P N P M = 0?Why?

5.13.14. Let M and N be subspaces of the same vector space, and let P M
and P N be orthogonal projectors onto M and N, respectively.
(a) Prove that R (P M + P N )= R (P M )+ R (P N )= M + N.
Hint: Use Exercise 4.2.9 along with (4.5.5).
(b) Explain why M⊥N if and only if P M P N = 0.
(c) Explain why P M + P N is an orthogonal projector if and only
if P M P N = 0, in which case R (P M + P N )= M⊕N and
M⊥N. Hint: Recall Exercise 5.9.17.

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5.13.15. Anderson–Duffin Formula. Prove that if M and N are subspaces
of the same vector space, then the orthogonal projector onto M∩N
is given by P M∩N =2P M (P M + P N ) P N . Hint: Use (5.13.12) and
†
Exercise 5.13.14 to show P M (P M + P N ) P N = P N (P M + P N ) P M .
†
†
†
Argue that if Z =2P M (P M +P N ) P M , then Z = P M∩N Z = P M∩N .
5.13.16. Given a square matrix X, the matrix exponential e X is defined as
∞
X 2 X 3 X n
X
e = I + X + + + ··· = .
2! 3! n!
n=0
It can be shown that this series converges for all X, and it is legitimate
to differentiate and integrate it term by term to produce the statements

de At /dt = Ae At =e At A and e At A dt =e At .
T
(a) Use the fact that lim t→∞ e −A At = 0 for all A ∈ m×n to
T
T
show A = ∞ −A At A dt.
e
†
0
k
e
(b) If lim t→∞ e −A k+1 t = 0, show A D = ∞ −A k+1 t A dt, where
0
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k = index(A).
(c) For nonsingular matrices, show that if lim t→∞ e −At = 0, then

e
A −1 = ∞ −At dt.
0
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W. N. Anderson, Jr., and R. J. Duffin discovered this formula for the orthogonal projector onto
†
an intersection in 1969. They called P M (P M + P N ) P N the parallel sum of P M and P N
because it is the matrix generalization of the scalar function r 1 r 2 /(r 1 + r 2 )= r 1 (r 1 + r 2 ) −1 r 2
that is the resistance of a circuit composed of two resistors r 1 and r 2 connected in parallel.
The simple elegance of the Anderson–Duffin formula makes it one of the innumerable little
sparkling facets in the jewel that is linear algebra.
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Amore useful integral representation for A D is given in Exercise 7.9.22 (p. 615).