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Unformatted text preview: 16 - 1 Matrix facts S. Lall, Stanford 2009.11.18.01 16 - Matrix facts
• Completion of squares • Block LDU matrix decomposition • Inverse of a block matrix • Inverse of a sum • Useful matrix identities • Push-through identity 16 - 2 Matrix facts S. Lall, Stanford 2009.11.18.01 Completion of Squares
the completion of squares formula for quadratic polynomials is b ax2 + 2bxy + dy 2 = a x + y a
2 b2 2 y + d− a when a > 0, this tells us the minimum with respect to x for ﬁxed y min ax + 2bxy + dy =
x∈R 2 2 b2 2 y d− a b which is achieved when x = − y . a this also gives a test for global positivity: ax + 2bxy + dy > 0 for all nonzero x, y ∈ R
2 2 ⇐⇒ b2 a > 0 and d − > 0 a 16 - 3 Matrix facts S. Lall, Stanford 2009.11.18.01 completion of squares for matrices
if A ∈ Rn×n and D ∈ Rm×m are symmetric matrices and B ∈ Rn×m, then x y
T AB BT D x = x + A−1By y T A x + A−1By + y T (D − B T A−1B )y • compare with b ax2 + 2bxy + dy 2 = a x + y a 2 b2 2 y + d− a • gives a general formula for quadratic optimization; if A > 0, then min
x x y T AB BT D x = y T (D − B T A−1B )y y and the minimizing x is xopt = −A−1By 16 - 4 Matrix facts S. Lall, Stanford 2009.11.18.01 block LDU matrix decomposition
the completion of squares formula gives a useful matrix decomposition x y
T AB BT D x = x + A−1By y x = y
T T A x + A−1By T + y T (D − B T A−1B )y I A−1B 0I x y I 0 B T A−1 I A 0 0 D − B T A−1B since this holds for all x, y , AB I 0 = BT D B T A−1 I holds whenever A is invertible A 0 0 D − B T A−1B I A−1B 0I 16 - 5 Matrix facts S. Lall, Stanford 2009.11.18.01 inverse of a block matrix
also holds for asymmetric matrices I 0 AB = CD CA−1 I A 0 0 D − CA−1B I A−1B 0I this decomposition is easy to invert AB CD
−1 I A−1B = 0I −1 A−1 0 0 (D − CA−1B )−1 A−1 0 0 (D − CA−1B )−1 −A−1BS −1 S −1 I 0 CA−1 I −1 I −A−1B = 0 I I 0 −CA−1 I A−1 + A−1BS −1CA−1 = −S −1CA−1 the matrix S = D − CA−1B is called the Schur complement of A 16 - 6 Matrix facts S. Lall, Stanford 2009.11.18.01 inverse of a sum
we can also complete the square to minimize w.r.t. y instead of x, which gives another formula, which holds whenever D is invertible AB CD
−1 T −1 = −D−1CT −1 −T −1BD−1 D−1 + D−1CT −1BD−1 where T = A − BD−1C AB the two formulas for CD −1 must be equal, so (A − BD−1C )−1 = A−1 + A−1B (D − CA−1B )−1CA−1 called the matrix inversion lemma or Sherman-Morrison-Woodbury formula 16 - 7 Matrix facts S. Lall, Stanford 2009.11.18.01 Useful Matrix Identities
A(I + A)−1 = I − (I + A)−1 because (I + A)(I + A)−1 = I so (I + A)−1 + A(I + A)−1 = I 16 - 8 Matrix facts S. Lall, Stanford 2009.11.18.01 more useful matrix identities
(I + AB )−1 = I − A(I + BA)−1B verify this directly; we have I − A(I + BA)−1B (I + AB ) = I + AB − A(I + BA)−1B (I + AB ) = I + AB − A(I + BA)−1(I + BA)B =I • I + AB is invertible if and only if I + BA is • true for any A and B , not just square ones 16 - 9 Matrix facts S. Lall, Stanford 2009.11.18.01 and more useful matrix identities
A(I + BA)−1 = (I + AB )−1A because B (I + AB ) = (I + BA)B • called push-through identity • A on the left pushes in, and pushes out A on the right ...
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.
- Fall '08