FAST QRS DECOMPOSITION OF MATRIX AND ITS APPLICATIONS TO NUMERICAL OPTIMIZATION JIAN-KANG ZHANG¤ AND KON MAX WONG Abstract. In this paper, we extend our recently developed equal diagonal QRS decomposition of a matrix to a general case in which the diagonal entries of the R-factor are not necessarily equal but satisfy a majority relationship with the singular values of the matrix. A low-complexity quadratic recursive algorithm to characterize all the eligible S-factors is derived. In particular, a pair of fast closed-form QRS decompositions for both a matrix and its inverse is obtained. Finally, we show how to apply this generalized QRS decomposition to e±ciently solving two classes of numerical opti- mization problems that usually occur in the joint design of transceiver pairs with decision feedback detection in communication systems via semi-de¯nite and geometrical programming. Key words. QR decomposition, majority, QRS decomposition, Q-R-S-factors, semi-de¯nite and geometrical programming 1. Introduction. The QR decomposition and the singular value decomposition (SVD) are commonly used tools in various signal processing applications. The QR decomposition of a matrix A is a factorization [1, 2] A = QR, where Q is a unitary matrix and R is an upper triangular matrix, while the SVD decomposition of an M £ N matrix A is a factorization [1, 2] µ ¶ ¤1=2 0 A = U r£(N¡r) VH ; (1.1) 0(M¡r)£r 0(M¡r)£(N¡r) where r is the rank of A, U and V are unitary and ¤ = diag(¸1; ¸2; ¢ ¢ ¢ ; ¸r) with ¸1 ¸ ¸2 ¸ ¢ ¢ ¢ ¸r > 0. To preserve the virtues of both decompositions, an intermediary between the QR decomposition and the QR decomposition, a two-sided orthogonal decomposition URV was developed in [2, 3, 4]; i.e., A = URVH , where U and V are unitary matrices and µ ¶ R 0 R = r£r r£(N¡r) (1.2) 0(M¡r)£r 0(M¡r)£(N¡r) with Rr£r being a r £ r upper triangular matrix. Recently, we have developed the equal diagonal QRS decomposition of a ma- trix [5, 6, 7, 8]; i.e., any matrix A can be decomposed into AS = QR, where Q and S are unitary matrices and R has the same structure as that given in (1.2) but Qr 1=2r diagonal entries of Rr£r are all equal to ( i=1 ¸i) . There are two main di®er- ences between the URV decomposition and the QRS decomposition. (a) Methodology is totally di®erent. In the URV decomposition, for a given data matrix A, perform trangularlization to the left corner by orthonormal row and column transformations. Such unitary matrices U and V always exist [2, 3, 4]. However, in our equal diagonal QRS decomposition, the existence of the Q and S factors is problematic in mathe- matics literature. (b) The task of designing the S-factor in the QRS decomposition is totally di®erent from that of designing the V-factor in the URV decomposition. The goal of designing the V-factor is rank-revealing and updating for subspace track- ing [2, 3, 4], while the principal purpose to design the S-factor is to optimize the ¤Contact Author: Department of Electrical and Computer Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S 4K1, Canada. Phone: +1 905 525 9140, Ext. 27599. Fax: +1 905 521 2922. ([email protected]). 1 2 J.-K. ZHANG AND K. M. WANG signal to noise (or plus interference) ratio of the worst case R-factor value subchannel or block error probability for decision feedback device in communication signal pro- cessing [5, 6, 7, 8]. Essentially, our equal diagonal QRS decomposition is based on the Schur's decomposition [1] that the determinant of a positive de¯nite matrix is equal to the product of the determinant of an arbitrarily given principal submatrix and the determinant of its schur complement. Therefore, it actually describes a process for successively and evenly distributing the total information over each one-dimensional subspace [8]. Very interestingly, such decomposition simultaneously enjoys many im- portant optimality properties in communication signal processing and therefore, has played a vital role in jointly designing transceiver with decision feedback detection in many block-based communication systems [5, 6, 7, 9]. However, in some environ- ments where di®erent R-factor value subchannels (users) require di®erent the qualities of service (Q±S) [10] or have di®erent space freedom such as random MIMO chan- nels [11, 12], this would demand an unequal diagonal QRS decomposition. Therefore, in this paper we extend the equal diagonal QRS decomposition [5, 6, 7] to a general case where the diagonal entries of the R-factor are not necessarily equal but satisfy a majority relationship [13, 14, 10]. A low-complexity generalized quadratic recursive algorithm to characterize all the S-factors is developed. In particular, a pair of fast speci¯c closed-form QRS decompositions is derived for both an invertible matrix and its inverse. Finally, we show how to apply this general QRS decomposition to e±- ciently solving two classes of numerical optimization problems that usually occur in the joint design of transceiver pairs with decision feedback detection in communication systems via semi-de¯nite and geometrical programming [15]. It should be mentioned that Mirsky [13] and Guess [10] gave their constructive proofs to ¯nd a speci¯c positive de¯nite matrix with a pair of the prescribed eigen- values and the Cholesky values. Although their methods can be exploited to ¯nd a unitary S-factor via the eigenvalue decomposition of the obtained positive de¯nite matrix, these are not direct and e±cient, since many applications require directly computing the S-factor. In addition, their methods [13, 10] cannot characterize all the eligible positive de¯nite matrices. Notation: Matrices are denoted by uppercase boldface characters (e.g., A), while column vectors are denoted by lowercase boldface characters (e.g., b). The (i; j)-th entry of A is denoted by Ai;j. The i-th entry of b is denoted by bi. The columns of an M £ N matrix A are denoted by a1; a2; ¢ ¢ ¢ ; aN . Notation Ak denotes a matrix consisting of the ¯rst k columns of A, i.e., Ak = [a1; a2; ¢ ¢ ¢ ; ak]. By convention, A0 = 1. The remaining matrix after deleting columns ak1 ; ak2 ; ¢ ¢ ¢ ; aki from A is denoted by Ak1;k2¢¢¢ ;ki . The j-th diagonal entry of a square matrix A is denoted ? by [A]j. Notation A denotes the orthonormal complement of a matrix A. The transpose of A is denoted by AT . The Hermitian transpose of A (i.e., the conjugate and transpose of A) is denoted by AH . 2. QRS decomposition of matrix. Our task in this section is to extend the equal diagonal QRS decomposition in [5, 6, 7, 8] and the quadratic recursive algorithm to a general case and then, derive a pair of a speci¯c closed-form QRS decompositions for both an invertible matrix and its inverse. 2.1. QRS decomposition. First, we give a quadratic recursive algorithm to determine whether a positive sequence is a valid candid for the diagonal entries of the R-factor in the QR decomposition of a given matrix. For discussion convenience, we assume hereafter that the singular value decomposition of an M £ N matrix A is given by (1.1). In order to judge whether a positive sequence constitutes a diagonal FAST QRS DECOMPOSITION OF MATRIX AND ITS APPLICATIONS 3 entries of the R-factor of a given matrix, we recall the notion of majorization and some key results [13, 16, 10] we will require in our derivation. K Definition 2.1. For any a 2 R , let a[1] ¸ a[2] ¸ ¢ ¢ ¢ ¸ a[K] denote the components of a in decreasing order (also termed order statistics of a). Majorization makes precise the vague notion that the components of a vector a are \less spread out" or \more nearly equal" than the components of a vector b. Definition 2.2. Let a; b 2 RK . Then, vector b majorizes a if Xk Xk a[i] · b[i] i=1 i=1 for all k = 1; 2; ¢ ¢ ¢ ;K, and with equality when k = K, which is denoted by a Á b. We also need the following de¯nition. K K Definition 2.3. Let fakgk=1 and fbkgk=1 be two positive real-valued sequences. K K It is said that fakgk=1 majorities fbkgk=1 in the product sense if Yk Yk a[i] ¸ b[i] (2.1) i=1 i=1 ­ for all k = 1; 2; ¢ ¢ ¢ ;K, and with equality when k = K, which is denoted by a > b. The following lemma [17, 16, 13, 10] gives a very simple necessary and su±cient condition to check whether a pair of positive sequences constitute a pair of the valid singular values and the R-factor values of some matrix, r r Lemma 2.4. Let f¸igk=1 and fdigi=1 be the eigenvalue sequence and the Cholesky r value sequence of a positive de¯nite matrix A, respectively. Then, f¸igi=1 majorities r r r fdigi=1 in the product sense. Conversely, if f¸igi=1 majorities fdkgi=1 in the product r sense, then, for an arbitrarily given desired permutation of fdigi=1, dj1 ; dj2 ; ¢ ¢ ¢ ; djr , r r there exists a matrix A such that f¸igi=1 and fdji gi=1 are the eigenvalues and the Cholesky values of A, respectively. In [13], Mirsky gave a constructive proof to ¯nd some positive de¯nite matrix with the given valid eigenvalues and the Cholesky values.