上海交通大学：《矩阵论》课程教学资源（讲义）CHAPTER 07

MATRIX THEORY CHAPTER Z FALL 2017 1.POSITIVE DEFINITE MATRIX Definition 1.An nx n Hermitian matri A is said to be positive definite if x*Ar>0 for all r∈Cn,x≠0： A is said to be positive semidifiniteif x'A≥0 for all∈C,x≠0. ()The characteristie polymomial p(t)=det(1)siics: The Theorem 2.Suppose A is an nxn Hermitian matriz.Then the following are equivalent: all (b)The matr. Theorem 3.If A is positive semidefinite,then for any positive integer k,there erists a unique B satisfying (a)B is positve semidefinite Hermitian; )B 宫m:甘A Theorem 4(Cholesky Decomposition).A matrir A is positive definite if and only if there erists a nonsigular lower triangular matriz L with positive diagonal entries such that A=LL.If A is real,L can be taken to be real. Example 1. [] .A is positive definite since det(5])>, )a(:)。 .A has a Cholesky decomposition A=LL'.Let .小-e 11

MATRIX THEORY - CHAPTER 7 FALL 2017 1. Positive Definite Matrix Definition 1. An n × n Hermitian matrix A is said to be positive definite if x ∗Ax > 0 for all x ∈ C n , x 6= 0; A is said to be positive semidifinite if x ∗Ax ≥ 0 for all x ∈ C n , x 6= 0. Theorem 1. Suppose A is an n × n Hermitian matrix. Then the following are equivalent: (a) A is positive definite. (b) The eigenvalues of A are all positive. (c) The characteristic polynomial pA(t) = det(tI − A) = t n + an−1t n−1 + · · · + a1t + a0 saitisfies: akak+1 0 where Ai is the submatrix of A determined by the first i rows and first i columns. (e) The exists a nonsingular matrix C such that A = C ∗C. Theorem 2. Suppose A is an n × n Hermitian matrix. Then the following are equivalent: (a) A is positive semidefinite. (b) The eigenvalues of A are all nonnegative. (b) The exists a matrix C such that A = C ∗C. Theorem 3. If A is positive semidefinite, then for any positive integer k, there exists a unique B satisfying (a) B is positve semidefinite Hermitian; (b) Bk = A; (c) AB = BA; (d) rank(B) = rank(A); (e) B is real symmetric if A is real symmetric. Theorem 4 (Cholesky Decomposition). A matrix A is positive definite if and only if there exists a nonsigular lower triangular matrix L with positive diagonal entries such that A = LL∗ . If A is real, L can be taken to be real. Example 1. A =   5 −1 3 −1 2 −2 3 −2 3   • A is positive definite since det ￾ 5
> 0, det  5 −1 −1 2  > 0, det     5 −1 3 −1 2 −2 3 −2 3     > 0 • A has a Cholesky decomposition A = LL∗ . Let L =   l11 l21 l22 l31 l32 l33   , L∗ =   l11 ¯l21 ¯l31 l22 ¯l32 l33   1

2 FALL 2017 where l>0.Then A=LL'gives the equations 12,=5 l2l=-1 41-3 l21L21+52=2 l121+l2l2=-2 lg11+l4232+保=3 So we solving out all the entries of L in the order

2 FALL 2017 where lii > 0. Then A = LL∗ gives the equations l 2 11 = 5 l21l11 = −1 l31l11 = 3 l21¯l21 + l 2 22 = 2 l31¯l21 + l32l22 = −2 l31¯l31 + l32¯l32 + l 2 33 = 3 So we solving out all the entries of L in the order l11, l21, l31, l22, l32, l32