13.21. EXERCISES 387

26. ↑Show that if L,L1 are lower triangular with ones down the main diagonal andH,H1 are upper triangular with all ones down the main diagonal and D, D1 are di-agonal matrices having all positive diagonal entries, and if LDH = L1D1H1, thenL = L1,H = H1,D = D1. Hint: Explain why D−1

1 L−11 LD = H1H−1. Then explain

why the right side is upper triangular and the left side is lower triangular. Concludethese are both diagonal matrices. However, there are all ones down the diagonal inthe expression on the right. Hence H = H1. Do something similar to conclude thatL = L1 and then that D = D1.

27. ↑Show that if A is a symmetric real matrix such that x→ ⟨Ax,x⟩ is positive def-inite, then there exists a lower triangular matrix L having all positive entries downthe diagonal such that A = LLT . Hint: From the above, A = LDH where L,H arerespectively lower and upper triangular having all ones down the diagonal and D is adiagonal matrix having all positive entries. Then argue from the above problem andsymmetry of A that H = LT . Now modify L by making it equal to LD1/2. This iscalled the Cholesky factorization.

28. Given F ∈ L (X ,Y ) where X ,Y are inner product spaces and dim(X) = n ≤ m =dim(Y ) , there exists R,U such that U is nonnegative and Hermitian (U = U∗) andR∗R = I such that F = RU. Show that U is actually unique and that R is determinedon U (X) . This was done in the book, but try to remember why this is so.

29. If A is a complex Hermitian n× n matrix which has all eigenvalues nonnegative,show that there exists a complex Hermitian matrix B such that BB = A.

30. ↑Suppose A,B are n×n real Hermitian matrices and they both have all nonnegativeeigenvalues. Show that det(A+B)≥ det(A)+det(B). Hint: Use the above problemand the Cauchy Binet theorem. Let P2 = A,Q2 = B where P,Q are Hermitian andnonnegative. Then

A+B =(

P Q)( P

Q

).

31. Suppose B =

(α c∗

b A

)is an (n+1)× (n+1) Hermitian nonnegative matrix

where α is a scalar and A is n×n. Show that α must be real, c= b, and A = A∗,Ais nonnegative, and that if α = 0, then b= 0. Otherwise, α > 0.

32. ↑If A is an n× n complex Hermitian and nonnegative matrix, show that there existsan upper triangular matrix B such that B∗B = A. Hint: Prove this by induction. It isobviously true if n = 1. Now if you have an (n+1)× (n+1) Hermitian nonnegative

matrix, then from the above problem, it is of the form

(α2 αb∗

αb A

),α real.

33. ↑ Suppose A is a nonnegative Hermitian matrix (all eigenvalues are nonnegative)which is partitioned as

A =

(A11 A12

A21 A22

)