Indeed in this case the proof is immediate: matrices of this form are nondegenerate, and the associated hermitian form admits an isotropic subspace of dim n. Since H is a subspace, it is implied by the question that the zero matrix satisfies the desired property. is a symmetric or Hermitian positive-definite matrix, all eigenvalues are positive. … positive definite matrix, A positive definite matrix has at least one matrix square root. This implies that all eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors. 2, pp. B>_ 0, a and 3are positive numbers unless Aand Bare both positive definite, in which case aand/can be any real numbers, andmis a positive integer. 4�����v�;�v����>�C���������h�'�!����¸�Sv�y@��{���>^~�����/���1���g���i���{�v|ao���,��X�p� �'���!�/tܘ��Q~�6��q��]���~��2d��F��a�6��k��s�p ʤ��mC������x��U*��v5� �ڐ�Z>� 3p�����c���u�j�;�u��B��*�i8Ԇl��u� ]n�p� �#��m8Ԇl��#���?X�vG��$V�W^��4d�-��[��1�5* ���hN���5r�l�M�m�]�A Dq���Ai�&�{}�+� N��ڣ�����5�|�O���ӿ��n���z��\*#�o����ʇ��K���'���Z��>h�0�C��1M�@Z �)$-l�����C�o������)�׃P�u���R�P��;>1|�����o�w#�m�7�Y��Z��[fd����av��@P�ܞ�\� =t�Br _8��J��Y� ,�c��Ife�Ajߌ���Čs8oy��"�q:_���c'��ծi9���y�S��=n�(�cB�)���V퐀�Q*Y�v�]{ip��@;��0��N�*��Cd}@)�Cyg2�ũ)f���no��ւ�������1��:��,��,�5�n�x�A�y�WU�! Positive definite matrix. The symbol denotes the spectral norm of the matrix . Inspired by our four definitions of matrix functions (diagonalization, Sylvester's formula, the resolvent method, and polynomial interpolation) that utilize mostly eigenvalues, we introduce a wide class of positive definite matrices that includes standard definitions used in mathematics. Since the determinant is a degree npolynomial in , this shows that any Mhas nreal or complex eigenvalues. This z will have a certain direction.. The values of λ that satisfy the equation are the generalized eigenvalues. EIGENVALUE INEQUALITIES 115 THEOREM 3. If A and B are positive … A close cousin is the positive semi-deﬁnite matrix. Let A be a Hermitian matrix of order n, and let B be a principal submatrix of A of order n−1.Ifλ Then, is an eigenvalue of Mif there is a non-zero vector ~vsuch that M~v= ~v This implies (M I)~v = 0, which also means the determinant of M I is zero. cations for algorithms that approximate eigenvalues of very large Hermitian matrix A with those of the small matrix H = Q∗AQ for some subunitary matrix Q ∈ n×m for m n. (In engineering applications n = 106 is common, and n = 109 22 2). Theorem: A Hermitian matrix A ∈ M n is positive semidefinite if and only if all of its eigenvalues are nonnegative. 2. Since the determinant is a degree npolynomial in , this shows that any Mhas nreal or complex eigenvalues. Thus, as a corollary of the problem we obtain the following fact: Eigenvalues of a real symmetric matrix are real. I preferred to formulate the question with k positive and k negative because it seems to me more intuitive (although equivalent). We still have that is positive semi-definite (definite) if and only if its eigenvalues are positive (resp. by Marco Taboga, PhD. 11.11. If Mz = λz (the defintion of eigenvalue), then z.TMz = z.Tλz = λ‖z²‖. have n positive and n negative eigenvalues. Thanks for the remark about the zero matrix. For people who don’t know the definition of Hermitian, it’s on the bottom of this page. 1 Basic properties of Hermitian matrices We recall that a matrix A2M nis called Hermitian if A = Aand skew-Hermitian if A = A, and we note that Ais Hermitian if and only if iAis skew-Hermitian. A necessary and sufficient condition for a complex matrix to be positive definite is that the Hermitian part A positive definite matrix has at least one matrix square root. In the following we denote byÂ Â the leading principal submatrix ofÂ Â determined by the firstÂ Â rows and columns:. The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. The info field indicates the location of (one of) the eigenvalue (s) which is (are) less than/equal to 0. the scalar product $(A,B)=\mathrm{tr}(AB)$), or equivalently there exists a basis of $\mathbb C^n$ such that all matrices in $H$ have zero trace. 6. Theorem 5.12. " Thus a matrix with a Cholesky decomposition does not imply the matrix is symmetric positive definite since it … 1. <> Denoted by {λ 1, λ 2, …, λ n} the spectrum of H(M) with decreasing ordering. Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues. principal submatrices of Hermitian matrices. C. A Hermitian matrix is diagonalisable if and only if its eigenvalues are positive. Then, is an eigenvalue of Mif there is a non-zero vector ~vsuch that M~v= ~v This implies (M I)~v = 0, which also means the determinant of M I is zero. The direction of z is transformed by M.. Thanks for contributing an answer to MathOverflow! A Hermitian (or symmetric) matrix is positive definite iff all its eigenvalues are positive. 8. In an 2n-dimensional space the space of block matrices of the form. semideﬁnite) matrix A. 8.2 Hermitian Matrices 273 Proof If v is a unit eigenvector of A associated with an eigenvalue λ, then Av = λv and vhA = vhAh = (Av)h = (λv)h = λ∗vh Premultiplying both sides of the ﬁrst equality by vh, postmultiplying both sides of the second equality by v, and noting that vhv = kvk2 = 1, we get vhAv = λ = λ∗ Hence all eigenvalues of A are real. . Hermitian matrices have real eigenvalues. Before giving veriﬁable characterizations of positive deﬁniteness (resp. Is there a good characterization of those $H$ such that every $A\in H$ has at least $k$ positive and $k$ negative eigenvalues? The generalized eigenvalue problem is to determine the solution to the equation Av = λBv, where A and B are n-by-n matrices, v is a column vector of length n, and λ is a scalar. < ik < n. HERMITIAN MATRICES, EIGENVALUE MULTIPLICITIES, AND EIGENVECTOR COMPONENTS∗ CHARLES R. JOHNSON† AND BRIAN D. SUTTON‡ SIAM J. MATRIX ANAL. Asking for help, clarification, or responding to other answers. Hermitian positive definite matrix. $\endgroup$ – Harald Hanche-Olsen Nov 9 '09 at 13:21 ... Eigenvalues of a sum of Hermitian positive definite circulant matrix and a positive diagonal matrix. There is a vector z.. A positive deﬁnite (resp. Unlike Dembo's bound the improved bound is always positive. It is well known [HH, Corollary 2.3] that the product of two positive semidefinite Hermitian matrices is diagonalizable andhas nonnegative eigenvalues. A complex-valued matrix Mis said to be Hermitian if for all i;j, we have M ij = M ji. APPL. The distribution function F ⁡ (s) given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of n × n Hermitian matrices; see Tracy and Widom . :��K�� r��)�W�|��������OԖq�=$�eV�ãJ�����9Z"/ jʊ���Wߠ"��}#�kԭ; ����GG�[̙���4�閬 ����9����B��� �� ��N������Ȉ���"��U13Aa%��e�7b����\clu�(ݗ;����b�3p 7�Z���� HK1�*.ܨ�'N�Á��>�,�U�O��(���L&����.lw ���@%��2�O;�� �w�4�/� m . (b) Eigenvectors for distinct eigenvalues of A are orthogonal. x��\K�G&8Np��p��[�~�P�1X��&B� ���س+ɒ�Ł�Nf=���gw��8��ʪ���e��|��������~x�b�|�c�7��ќ����y��r���������{7I�_h�ϧ�������O����o6,��s��������ݨ'��ۋ���\����*9߾܉I%�����'c���QN�Y���v��Ny�����uK>�ds��N�£�Bnym���v���a����/�y�gu��Ǐ�ǰ������*鐰q�nܨ %�쏢 If the 2 n] n n ∈ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗} ∗ ∗. Since z.TMz > 0, and ‖z²‖ > 0, eigenvalues (λ) must be greater than 0! Even for$k=2$I was not able to find any good characterization. The same is true of any symmetric real matrix. As for any positive matrix, ifÂ Â is positive definite, thenÂ allÂ principal minors ofÂ Â are positive; whenÂ Â isÂ Hermitian… Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 0) for all x2Cn nf0g: We write A˜0 (resp.A 0) to designate a positive deﬁnite (resp. Start with A TAx D x. The matrix is a partial Hermitian matrix with a chordal graph. The first condition implies, in particular, that , which also follows from the second condition since the determinant is the product of the eigenvalues. where C1=2 denotes the Hermitian positive deﬁnite square root of the Her-mitian positive deﬁnite matrix C. Hence all the eigenvalues are real. A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the "i"th row and "j"th column is equal to the complex conjugate of the element in the "j"th row and "i"th column, for all indices "i" and "j":: a_{i,j} = overline{a_{j,i. The determinant of a positive definite matrix is always positive, so a positive definite matrix is always nonsingular. The eigenvalues are 1-t and 1+t with the corresponding eigenvectors {-i, 1} and {i, 1}. The characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity. Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues. This z will have a certain direction.. Let$H$be a linear subspace of the space of Hermitian$n\times n$matrices. In the following we denote by A i the leading principal submatrix of A determined by the first i rows and columns: A i ≡ A ({1, 2, …, i }), i = 2, …, n . 7 0 obj Let A be a complex Hermitian matrix. We say that A is also positive deﬁnite if for every non-zero x ∈CN, xHAx > 0. ���2�n�7��lsBS &mH��ZBP� ʋ�c߯��h��q���/0�x��ԭH�q7�Ih!��q����׵�3�O��{�))�u����?H�\��7A��f �ۻh���XkT7�h�d�\5S�A��ɸs.+��-�+����x+��X�/PP��+ �%k4��� E���|^}_@��7��U-e��̽2�N�"2�f���� �l�d�:}X/A۳*m��8h��(�}��R�L"q�Y 0��P���B) )���h����@��Sװ�y&���"9��!����)���{��\���%��F �k�. Example: Find the eigenvalues and eigenvectors of the real symmetric (special case of Hermitian) matrix below. A Hermitian (symmetric) matrix with all positive … If and are positive definite, then so is . In this paper we consider the behaviour of a real eigenvalue of an analytic Hermitian matrix valued function under perturbation with a positive semidefinite analytic Hermitian matrix valued function. ("z.T" is z transpose. However most eigensolvers sort them by increasing eigenvalues exchanging the eigenvectors from negative to positive t (see code and output plot). It is positive definite if and only if all of its eigenvalues are positive. For people who don’t know the definition of Hermitian, it’s on the bottom of this page. 26, No. Notice that this is a block diagonal matrix, consisting of a 2x2 and a 1x1. A Hermitian matrix is positive definite if and only if its eigenvalues are all positive: The eigenvalues of m are all positive: A real is positive definite if and only if its symmetric part, , is positive definite: I will show now that the eigenvalues of ATA are positive, if A has independent columns. (a) Suppose λ is an eigenvalue … Show < q,z >:= zHAq deﬁnes a valid inner product on Cn. The eigenvalues of a Hermitian matrix are non-real. GENERAL H−MATRICES, http://www.emis.de/journals/ELA/ela-articles/articles/vol18_pp801-820.pdf. To see why this relationship holds, start with the eigenvector equation Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Hence sorting according to the "eigenvector character" the eigenvalues should cross at t = 0. Hermitian matrices can be with non-real complex entries and have real eigenvalues which may be negative also. The page says " If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero. rev 2020.12.3.38123, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. More-over, if H ‹Qdiag–kiƒQ is a spectral decomposition then XBX ‹I and XAX ‹diag–k iƒ, where X ‹Bÿ1=2Q, that is, A and B are simultaneously The determinant of a positive definite matrix is always positive, so a positive definitematrix is always nonsingular. Its only principal subma- trix is the positive semidefinite 1 x 1 matrix (0). ∴ A Positive Definite Matrix must have positive eigenvalues. Furthermore, exactly one of its matrix square roots is itself positive definite. Here are some other important properties of symmetric positive definite matrices. Unitary and hermitian matrices 469 Proposition 11.107: Eigenvalues and eigenvectors of hermitian matrices Let A be a hermitian matrix. Let A be HPD. > Featured on Meta Feature Preview: Table Support Positive semi-definite matrices are real symmetric and have non-negative eigenvalues. We have observed earlier that the diagonal entries of a Hermitian matrix are real. Sponsored Links EXAMPLE 1.1. If and are positive definite, then so is . This can also be viewed as a Obviously the Hermitian adjacency matrix is Hermitian and all eigenvalues are real. Abstract: Presents an improvement to Demho's (1988) lower bound on the smallest eigenvalue of a Hermitian positive-definite matrix. Moreover, since is Hermitian, it is normal and its eigenvalues are real. When we multiply matrix M with z, z no longer points in the same direction. A useful consequence of HPD (SPD) matrices is that their eigenvalues (which wealreadyknowarerealdue to the Hermitianproperty)must be NON-NEGATIVE. D. Every Hermitian matrix is diagonalisable Reset Selection search TE AN W Part 1 of 1 Question 2 of 13 4 Points Let U, and U2 be unitary matrices. A complex-valued matrix Mis said to be Hermitian if for all i;j, we have M ij = M ji. The entries in the diagonal matrix † are the square roots of the eigenvalues. Some Inequalities for the Eigenvalues of the Product of Positive Semidefinite Hermitian Matrices Boying Wang and Fuzhen Zhang* Department of Mathematics Beijing Normal University Beijing, People's Republic of China Submitted by George P. H. Sty an ABSTRACT Let A1(A) > > AA) denote the eigenvalues of a Hermitian n by n matrix A, and let 1 _< il < . For example, if a matrix has an eigenvalue on the order of eps, then using the comparison isposdef = all(d > 0) returns true, even though the eigenvalue is numerically zero and the matrix is better classified as symmetric positive semi-definite. If A is equal to its conjugate transpose, or equivalently if A is Hermitian, then every eigenvalue is real. Also in the complex case, a positive definite matrix is full-rank (the proof above remains virtually unchanged). Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues. Returns two objects, a 1-D array containing the eigenvalues of a, and a 2-D square array or matrix (depending on the input type) of the corresponding eigenvectors (in columns). (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. B. If an Hermitian positive definite matrix satisfies , we denote by . This implies that the eigenvalues of any induced subdigraph interlace those of the digraph itself. Since is Hermitian, it has an eigendecomposition = − where is unitary and is a diagonal matrix whose entries are the eigenvalues of Since is positive semidefinite, the eigenvalues are non-negative real numbers, so one can define as the diagonal matrix whose … is a symmetric or Hermitian positive-definite matrix, all eigenvalues are positive. OF NONSTRICTLY DIAGONALLY DOMINANT MATRICES AND Hermitian matrices with prescribed number of positive and negative eigenvalues, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, matrices self-adjoint with respect to some inner product, Eigenvalues of a sum of Hermitian positive definite circulant matrix and a positive diagonal matrix, The eigenvectors and eigenvalues of matrix geometric mean, Upper bound for sum of absolute values of eigenvalues of Hermitian matrix, Parametrization of positive semidefinite matrices. ), THE EIGENVALUE DISTRIBUTION OF SCHUR COMPLEMENTS A Hermitian matrix is not always diagonalisable. The matrices AAT and ATA have the same nonzero eigenvalues. A matrix $$A \in \C^{n \times n}$$ is Hermitian positive definite (HPD) if and only if it is Hermitian ($$A^H = A$$) and for all nonzero vectors $$x \in \C^n$$ it is the case that $$x ^H A x \gt 0 \text{. Use MathJax to format equations. Section 4.2 Properties of Hermitian Matrices. Making statements based on opinion; back them up with references or personal experience. We know that its eigenvalues { λ 0, …, λ n − 1 } are extraordinarily nice in that they are positive reals and are the output of the discrete Fourier transform of the top row of C. Consider the sum C + D where D = diag MathOverflow is a question and answer site for professional mathematicians. Then (a) All eigenvalues of A are real. It is positive definite if and only if all of its eigenvalues are positive. Suppose I have a real n × n matrix C that is Hermitian, positive-definite, and circulant. x ] ≥ 0 is satisfied: The eigenvalues of are all non-negative: The usual version of the eigenvalue interlacing property states that the eigenvalues of any principal submatrix of a Hermitian matrix interlace those of the whole matrix (see [8, Theorems 4.3.8 and 4.3.15]). 4. It only takes a minute to sign up. For k=1 a nice characterization is the following: there is a positive definite matrix B orthogonal to H (w.r.t. Therefore, HPD (SPD) matrices MUST BE INVERTIBLE! 390–399 Abstract. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. When we multiply matrix M with z, z no longer points in the same direction. semideﬁniteness), we Az = λ z (or, equivalently, z H A = λ z H).. Theorem 1 (Cauchy Interlace Theorem). import numpy as np If is Hermitian positive definite (semidefinite), then we write . Nitpick: You need to restrict attention to the nonzero matrices in H. And a trivial observation: Since H is closed under negation you only need a condition that ensures at least k positive eigenvalues. We have observed earlier that the diagonal entries of a Hermitian matrix are real. They are plus or minus the singular values of A. 1 Basic properties of Hermitian matrices We recall that a matrix A2M nis called Hermitian if A = Aand skew-Hermitian if A = A, and we note that Ais Hermitian if and only if iAis skew-Hermitian. There is a vector z.. m . The direction of z is transformed by M.. A Hermitian matrix is positive semidefinite if and only if its eigenvalues are all non-negative: The condition Re [ Conjugate [ x ] . (Meaning eigenvalues of |A|=(A*A)^(1/2)). Proof. %PDF-1.4 The determinant of a positive definite matrix is usually positive, so a positive definite matrix is usually nonsingular. This matrix was introduced by Liu and Li and independently by Guo and Mohar . Positive definite symmetric matrices have the property that all their eigenvalues are positive. eigenvalues of a set of Hermitian matrices with positive semideﬁnite sum of bounded rank. This answers a question of A. Barvinok. numpy.linalg.eigh¶ linalg.eigh (a, UPLO='L') [source] ¶ Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix. In the following we denote byÂ Â the leading principal submatrix ofÂ Â determined by the firstÂ Â rows and columns:. This section serves a preparatory role for the next section---roots (mostly square). And a trivial observation: Since H is closed under negation you only need a condition that ensures at least k positive eigenvalues. strictly positive) real numbers. Hermitian matrices can be understood as the complex extension of real symmetric matrices. The Cauchy interlace theorem states that the eigenvalues of a Hermitian matrixAof ordernare interlaced with those of any principal submatrix of ordern−1. We use to denote all eigenvalues (each repeated as many times as its algebraic multiplicity) of an Hermitian matrix . [V,D,W] = eig(A,B) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'*B. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. To learn more, see our tips on writing great answers. First of all, the eigenvalues must be real! To subscribe to this RSS feed, copy and paste this URL into your RSS reader. A square matrix is positive definite if pre-multiplying and post-multiplying it by the same vector always gives a positive number as a result, independently of how we choose the vector.. (This fact is in Bhatia's matrix analysis book. stream By the previous theorem, this matrix is SPD. principal submatrices of Hermitian matrices. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. We will denote by λ i (M) the jth largest eigenvalue of H(M) (multiplicities counted). A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. As for any positive matrix, ifÂ Â is positive definite, thenÂ allÂ principal minors ofÂ Â are positive; whenÂ Â isÂ Hermitian… Corollary Every real symmetric matrix is Hermitian. This can also be viewed as a . Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues. 4. Hermitian Positive Definite Solution of (a) Prove that the eigenvalues of a real symmetric positive-definite matrix Aare all positive. The eigenvalues and eigenvectors of Hermitian matrices have some special properties. 1 Key words and phrases: Majorization, positive semideﬁnite 2 Abstract Let H = M K K∗ N 3 be a Hermitian matrix. But F(x) has both a positive eigenvalue and a negative eigenvalue, for all real values of x. x ] ≥ 0 is satisfied: The eigenvalues of are all non-negative: And a trivial observation: Since H is closed under negation you only need a condition that ensures at least k positive eigenvalues. It--it completely displays the eigenvalues and eigenvectors the symmetry of the--of the whole thing, because--that product, Q times lambda times Q transpose, if I transpose it, it--this comes in this position and we get that matrix back again. }$$ If in addition $$A \in \R^{n \times n}$$ then $$A$$ is said to be symmetric positive … If A is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.$\endgroup\$ – Harald Hanche-Olsen Nov 9 '09 at 13:21 ... Eigenvalues of a sum of Hermitian positive definite circulant matrix and a positive diagonal matrix. semideﬁnite) matrix is a Hermitian matrix A2M n satisfying hAx;xi>0 (resp. Exercise 5.2. To solve a symmetric eigenvalue problem with LAPACK, you usually need to reduce the matrix to tridiagonal form and then solve the eigenvalue problem with the tridiagonal matrix obtained. Symmetric eigenvalue problems are posed as follows: given an n-by-n real symmetric or complex Hermitian matrix A, find the eigenvalues λ and the corresponding eigenvectors z that satisfy the equation. Deﬁnition 5.14. the eigenvalues of are all positive. Compute the eigenvalues, observe they are both positive. To solve a symmetric eigenvalue problem with LAPACK, you usually need to reduce the matrix to tridiagonal form and then solve the eigenvalue problem with the tridiagonal matrix obtained. Only if A is nondegenerate. The proofs are almost identical to those we have seen for the real case. The same is true of any symmetric real matrix. c 2004 Society for Industrial and Applied Mathematics Vol. If the Browse other questions tagged stochastic-processes eigenvalues-eigenvectors markov-chains positive-semidefinite or ask your own question. Section 6.5 showed that the eigenvectors of these symmetric matrices are orthogonal. MathJax reference. Hence, there is no positive semidefinite completion.’ The eigenvalue for the 1x1 is 3 = 3 and the normalized eigenvector is (c 11) =(1). A Hermitian matrix is positive semidefinite if and only if its eigenvalues are all non-negative: The condition Re [ Conjugate [ x ] . It is positive definite if and only if all of its eigenvalues are positive. Some linear algebra functions and factorizations are only applicable to positive definite matrices. We extend previous results on perturbation with positive definite functions to the positive semidefinite case. 7. Since H-1 and H1/2GH1/2 are both positive definite Hermitian, we have by Theorem 1 k k F1Ai, (G) FjAi (H 1/2GH1/2) t=1 t=1 k = 1-1 Ai (H IH112GH1/2) t=1 k < rlAi (H 1)At (H1/2GH1/2) t=1 k k = II Ai (H-1)At (GH) = II An-i +1 (H)At (GH), t=1 't=1 Hence the theorem follows. Exception thrown when the input matrix was not positive definite.