Fachbereich Mathematik & Informatik
Freie Universität Berlin
Prof. Dr. Carsten Gräser, Tobias Kies
12th exercise for the lecture
Numerics IV
Winter Term 2016/2017
http://numerik.mi.fu-berlin.de/wiki/WS_2016/NumericsIV.php
Due: Tuesday, Jan 31st, 2017
Exercise 1 (4 TP)
For M ∈ Rm×n the Moore–Penrose pseudoinverse M + ∈ Rn×m is the matrix which is uniquely
determined by the properties
M M + M = M,
M +M M + = M +,
(M M + )T = M M + ,
(M + M )T = M + M.
For M ∈ Rn×n and an index set I ⊂ {1, . . . , n} we denote by MI ∈ Rn×n the matrix with
(MI )i,j = Mi,j for i, j ∈ I and (MI )i,j = 0 else. Now let A ∈ Rn×n and I ⊂ {1, . . . , n}.
a) Show that AI = II AII
b) Assume that the |I| × |I| submatrix of A indexed by I is regular. Show that (AI )+ =
(AI + I − II )−1 − I + II .
Exercise 2 (4 TP)
Let A ∈ Rm×n .
a) Suppose A is surjective. Show that A+ = AT (AAT )−1 .
b) Suppose A is injective. Show that A+ = (AT A)−1 AT .
c) Suppose A is diagonal. Show that (A+ )ij =
1
aij
if aij 6= 0 and (A+ )ij = 0 if aij = 0.
d) Suppose A is symmetric with eigenvalue decomposition A = QDQT where D is diagonal
and Q is orthonormal. Prove A+ = QD+ QT .
Please turn over...
Exercise 3 (4 TP)
Let A ∈ Rn×n be symmetric and B ∈ Rm×n .
a) Show that there exists λ0 ∈ R>0 such that A+λI is invertible for λ ∈ (−λ0 , λ0 )\{0} =: Λ.
b) Prove that the limits
P := lim (A + λI)−1 A = lim A(A + λI)−1
Λ3λ→0
Λ3λ→0
exist and coincide and that P is the Euclidean projection onto im(A) := {Ax | x ∈ Rn }.
Hint: Use the orthogonal decomposition Rn = im(A) + ker(A) and the eigenvalue decomposition of A.
c) Show that the limits
B := lim (B T B + λ2 In )−1 B T = lim B T (BB T + λ2 Im )−1
λ→0
λ→0
exist and coincide and that BB is the Euclidean projection onto im(B T B).
Hint: Use the orthogonal decomposition Rm = im(B) + ker(B T ).
Remark: From this it is possible to show that B = B + holds. Thus above limits give
an alternative characterization of the Moore-Penrose pseudoinverse.
Exercise 4 (4 TP)
Let M ∈ Rn×n symmetric positive definite and let M (k) ∈ Rn×n be a sequence of symmetric
positive definite matrices such that M (k) → M for k → ∞. Suppose A ⊆ {1, . . . , n} and let
I := {1, . . . , n} \ A. Prove that for any sequence (αk )k∈N ⊆ R>0 with αk → ∞ for k → ∞
holds
lim (M (k) + αk IA )−1 = (MI )+ .
k→∞
Here (IA )ij = δij for i ∈ A and (IA )ij = 0 for i ∈ I is the reduced identity matrix that only
contains the entries associated to the index set A, and similary (MI )ij = Mij for i, j ∈ I and
(MI )ij = 0 else.
A B
∈ R(n+m)×(n+m) with A ∈ Rn×n
Hint: You can use without proof that for a matrix
C D
invertible its Schur complement is given by S = D − CA−1 B, and that
−1 −1
In −A−1 B
A
0
In
0
A B
.
=
0
Im
0
S −1
−CA−1 Im
C D
Use the Schur complement applied to a suitable splitting of the above matrices in order to
compute the limit of the inverses.
Have fun!