AN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT


 Garry Haynes
 1 years ago
 Views:
Transcription
1 Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp AN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT J. CAI Communicated by Mohammad Asadzadeh Abstract. In this paper, an iterative method is proposed for solving the matrix inverse problem AX = B for Hermitiangeneralized Hamiltonian matrices with a submatrix constraint. By this iterative method, for any initial matrix A 0, a solution A can be obtained in finite iteration steps in the absence of roundoff errors, and the solution with least norm can be obtained by choosing a special kind of initial matrix. Furthermore, in the solution set of the above problem, the unique optimal approximation solution to a given matrix can also be obtained. A numerical example is presented to show the efficiency of the proposed algorithm. 1. Introduction Thought this paper, we adopt the following notation. Let C m n (R m n ) and HC n n denote the set of m n complex (real) matrices and n n Hermitian matrices, respectively. For a matrix A C m n, we denote its conjugate transpose, transpose, trace, column space, null space and Frobenius norm by A H, A T, tr(a), R(A), N(A) and A, respectively. In space C m n, we define inner product as: A, B = tr(b H A) for all MSC(2010): Primary: 15A29; Secondary: 65J22. Keywords: Inverse problem, Hermitiangeneralized Hamiltonian matrix, submatrix constraint, optimal approximation. Received: 7 August 2012, Accepted: 30 November c 2013 Iranian Mathematical Society. 129
2 1250 Cai A, B C m n, and the symbol Re A, B and A, B stand for its real part and conjugate number, respectively. Two matrices A and B are orthogonal if A, B = 0. Let Q s,n = {a = (a 1, a 2,, a s ) : 1 a 1 < a 2 < < a s n} denote the strictly increasing sequences of s elements from 1, 2,, n. For A C m n, p Q s,m and q Q t,n, let A[p q] stand for the s t submatrix of A determined by rows indexed by p and columns indexed by q. Let I n = (e 1, e 2,, e n ) be the n n unit matrix, where e i denotes its ith column. Let J n R n n be the orthogonal skewsymmetric matrix, i.e., Jn T J n = J n Jn T = I n and Jn T = J n. A matrix A C n n is called Hermitiangeneralized Hamiltonian if A H = A and (AJ n ) H = AJ n. The set of all n n Hermitiangeneralized ( Hamiltonian ) matrices is denoted 0 by HGH n n Ik. Particularly, if J n =, then the set HGH I k 0 n n reduces to the wellknown set of HermitianHamiltonian matrices, which have applications in many areas such as linearquadratic control problem [?,?], H optimization [?] and the related problem of solving algebraic Riccati equations [?]. Recently, there have been several papers considering solving the inverse problem AX = B for various matrices by direct methods based on different matrix decompositions. For instance, Xu and Li [?], Peng [?] and Zhou et al. [?] discuss its Hermitian reflexive, antireflexive solutions and leastsquare centrosymmetric solutions, respectively. Then Huang and Yin [?] and Huang et al. [?] generalize the results of the latter to the more general Rsymmetric and Rskew symmetric matrices, respectively. Li et al. [?] consider the inverse problem for symmetric P symmetric matrices with a submatrix constraint. Peng et al. [?,?] and Gong et al. [?] consider solving the inverse problem for centrosymmetric, bisymmetric and HermitianHamiltonian matrices, respectively, under the leading principal submatrix constraint. Zhao et al. [?] concerns the inverse problem for bisymmetric matrices under a central principal submatrix constraint. However, the inverse problem for the Hermitiangeneralized Hamiltonian matrices with general submatrix constraint has not been studied till now. Hence, in this paper, we consider solving the following problem and its associated best approximation problem which occurs frequently in experimental design ([?,?,?,?]) by iterative methods.
3 Iterative Hermitiangeneralized Hamiltonian solutions 1251 Problem I. Given X, B C n m, A S HC s t, p = (p 1, p 2,, p s ) Q s,n, and q = (q 1, q 2,, q t ) Q t,n, find A HGH n n, such that (1.1) AX = B and A[p q] = A S. Problem II. Let S E denote the set of solutions of Problem I, for given Ā Cn n, find Â S E, such that (1.2) Â Ā = min A Ā. A S E The rest of this paper is organized as follows. In Section 2, we propose an iterative algorithm for solving Problem I and present some basic properties of this algorithm. In Section 3, we consider the iterative method for solving Problem II. A numerical example is given in Section to show the efficiency of the proposed algorithm. Conclusions will be put in Section Iterative algorithm for solving Problem I Firstly, we present several basic properties of Hermitiangeneralized Hamiltonian matrices in the following lemmas. Lemma 2.1. Consider a matrix Y C n n. Then Y + Y H + J n (Y + Y H )J n HGH n n. Proof. The proof is easy, thus is omitted. Lemma 2.2. Suppose a matrix Y C n n and a matrix D HGH n n. Then Re Y, D = Y + Y H + J n (Y + Y H )J n, D. Proof. Since Y H, D = tr(d H Y H ) = tr((y D) H ) = Y, D H = Y, D, we have Y + Y H, D = 2Re Y, D. Then we get J n (Y + Y H )J n, D = tr(d H J n (Y + Y H )J n ) = tr(j n D H J n (Y + Y H )) = tr(d H (Y + Y H )) = Y + Y H, D = 2Re Y, D. Hence we have Y + Y H + J n (Y + Y H )J n, D = Re Y, D. Next we propose an iterative algorithm for solving Problem I.
4 1252 Cai Algorithm 1. Step 1. Input X, B C n m, A S HC s t, p = (p 1, p 2,, p s ) Q s,n, q = (q 1, q 2,, q t ) Q t,n and an arbitrary A 0 HGH n n ; Step 2. Compute R 0 = B A 0 X; S 0 = A S (e p1, e p2,, e ps ) T A 0 (e q1, e q2,, e qt ); E 0 = R 0 X H + (e p1, e p2,, e ps )S 0 (e q1, e q2,, e qt ) T ; F 0 = 1 [E 0 + E H 0 + J n (E 0 + E H 0 )J n ]; P 0 = F 0 ; k := 0; Step 3. If R k = S k = 0 then stop; else, k := k + 1; Step. Compute α k 1 = R k S k 1 2 P k 1 2 ; A k = A k 1 + α k 1 P k 1 ; R k = B A k X; S k = A S (e p1, e p2,, e ps ) T A k (e q1, e q2,, e qt ); E k = R k X H + (e p1, e p2,, e ps )S k (e q1, e q2,, e qt ) T ; F k = 1 [E k + E H k β k 1 = tr(f kp k 1 ) P k 1 2 ; P k = F k β k 1 P k 1 ; Step 5. Go to Step 3. + J n(e k + E H k )J n]; Remark 2.3. By Lemma 2.1, one can easily see that the matrix sequences {A k },{P k } and {F k } generated by Algorithm 1 are all the Hermitiangeneralized Hamiltonian matrices. And Algorithm 1 implies that if R k = S k = 0, then A k is the solution of Problem I. We list some basic properties of Algorithm 1 as follows. Theorem 2.. Assume that A is a solution of Problem I. Then the sequences {A i }, {P i }, {R i } and {S i } generated by Algorithm 2.1 satisfy the following equality: (2.1) P i, A A i = R i 2 + S i 2, i = 0, 1, 2,. Proof. From Remark 2.3, it follows that A A i HGH n n, i = 0, 1, 2,. Then according to Lemma 2.2 and Algorithm 1, for i = 0, we have P 0, A A 0 = 1 (E 0 + E H 0 + J n (E 0 + E H 0 )J n ), A A 0 = Re E 0, A A 0
5 Iterative Hermitiangeneralized Hamiltonian solutions 1253 = Re R 0 X H + (e p1, e p2,, e ps )S 0 (e q1, e q2,, e qt ) T, A A 0 = Re (e p1, e p2,, e ps )S 0 (e q1, e q2,, e qt ) T, A A 0 +Re R 0 X H, A A 0 = Retr((e q1, e q2,, e qt ) T (A A 0 )(e p1, e p2,, e ps )S 0 ) +Retr(X H (A A 0 )R 0 ) = Retr(R H 0 R 0) + Retr(S H 0 S 0) = tr(r H 0 R 0) + tr(s H 0 S 0) = R S 0 2. Assume that the conclusion holds for i = k(k > 0), i.e., P k, A A k = R k 2 + S k 2, then for i = k + 1, we have P k+1, A A k+1 = F k+1, A A k+1 β k P k, A A k+1 = 1 E k+1 + E H k+1 + J n(e k+1 + E H k+1 )J n, A A k+1 β k P k, A A k α k P k = Re E k+1, A A k+1 β k P k, A A k + β k α k P k 2 = R k+1 X H + (e p1, e p2,, e ps )S k+1 (e q1, e q2,, e qt ) T, A A k+1 β k ( R k 2 + S k 2 ) + β k R k 2 + S k 2 P k 2 P k 2 = R k S k+1 2. This completes the proof by the principle of induction. Remark 2.5. Theorem 2. implies that if Problem I is consistent, then R i 2 + S i 2 0 implies that P i 0. On the other hand, if there exists a positive number k such that R k 2 + S k 2 0 but P k = 0, then Problem I must be inconsistent. Lemma 2.6. For the ( sequences ) {R i }, {S i (}, {P i }) and {F i } generated by Algorithm 1, let ˆR Ri Si i = and Ŝi =. Then it follows that R H i (2.2) ˆR i+1, ˆR j + Ŝi+1, Ŝj = ˆR i, ˆR j + Ŝi, Ŝj 2α i F j, P i. Proof. By Algorithm 1, Remark 2.3 and Lemma 2.2, we have ˆR i+1, ˆR j + Ŝi+1, Ŝj = tr(r H j R i+1 +R j R H i+1)+ tr(s H j S i+1 + S j S H i+1) = tr(r H j (R i α i P i X) + R j (R i α i P i X) H ) + tr(s H j (S i α i (e p1, e p2,, e ps ) T P i (e q1, e q2,, e qt )) + S j (S i α i (e q1, e q2,, e qt ) T P i (e p1, e p2,, e ps ))) S H i
6 125 Cai = tr(r H j R i + R j R H i ) + tr(sh j S i + S j S H i ) α i tr(r H j P ix + R j (P i X) H +S H j (e p 1, e p2,, e ps ) T P i (e q1, e q2,, e qt ) + S j (e q1, e q2,, e qt ) T P i (e p1, e p2,, e ps )) = ˆR i, ˆR j + Ŝi, Ŝj α i E i + E H i, P i = ˆR i, ˆR j + Ŝi, Ŝj α i 2 E i + E H i + J n (E i + E H i )J n, P i = ˆR i, ˆR j + Ŝi, Ŝj 2α i F i, P i. Theorem 2.7. For the sequences { ˆR i }, {Ŝi} and {P i } generated by Algorithm 1, if there exists a positive number k such that ˆR i 0 for all i = 0, 1, 2,..., k, then we have (2.3) ˆR i, ˆR j + Ŝi, Ŝj = 0, (i, j = 0, 1, 2,..., k, i j). Proof. Since ˆR i, ˆR j = ˆR j, ˆR i and Ŝi, Ŝj = Ŝj, Ŝi, we only need to prove that (2.3) holds for all 0 j < i k. For k = 1, it follows from Lemma 2.6 that ˆR 1, ˆR 0 + Ŝ1, Ŝ0 = ˆR 0, ˆR 0 + Ŝ0, Ŝ0 2α 0 F 0, P 0 = tr(r H 0 R 0 + R 0 R H 0 ) + tr(sh 0 S 0 + S 0 S H 0 ) 2 R S 0 2 P 0 2 P 0, P 0 = 2( R S 0 2 ) 2 R S 0 2 P 0 2 P 0 2 = 0. and P 1, P 0 = F 1 tr(f 1P 0 ) P 0 2 P 0, P 0 = 0, Assume that ˆR m, ˆR j + Ŝm, Ŝj = 0 and P m, P j = 0 hold for all 0 j < m, 0 < m k. We shall show that ˆR m+1, ˆR j + Ŝm+1, Ŝj = 0 and P m+1, P j = 0 hold for all 0 j < m + 1, 0 < m + 1 k. For 0 j < m, by Lemma 2.6, we have ˆR m+1, ˆR j + Ŝm+1, Ŝj = ˆR m, ˆR j + Ŝm, Ŝj 2α m F j, P m = ˆR m, ˆR j + Ŝm, Ŝj α m P j + β j 1 P j 1, P m = α m P j, P m = 0, and P m+1, P j = F m+1, P j β m P m, P j = F m+1, P j = ˆR j, ˆR m+1 + Ŝj, Ŝm+1 + ˆR j+1, ˆR m+1 + Ŝj+1, Ŝm+1 2α m+1 = 0.
7 Iterative Hermitiangeneralized Hamiltonian solutions 1255 For j = m, it follows from Lemma 2.6 and the hypothesis that ˆR m+1, ˆR m + Ŝm+1, Ŝm = ˆR m, ˆR m + Ŝm, Ŝm 2α s F m, P m = 2( R m 2 + S m 2 ) 2α s P m + β m 1 P m 1, P m = 2( R m 2 + S m 2 ) 2 R m 2 + S m 2 P m 2 P m, P m = 0, and P m+1, P m = F m+1 β m P m, P m = F m+1, P m tr(f m+1p m ) P m 2 P m 2 = 0. Hence ˆR m+1, ˆR j + Ŝm+1, Ŝj = 0 and P m+1, P j = 0 hold for all 0 j < m + 1, 0 < m + 1 k. This completes the proof by the principle of induction. Remark 2.8. Based on Theorem 2.7, we can further ( demonstrate ) the ˆRk finite termination property of Algorithm 1. Let Z k =. Theorem 2.7 implies that the matrix sequences Z 0, Z 1, are orthogonal to each other in the finite dimension matrix subspace. Hence there exists a positive integer t 0 such that Z t0 = 0. Then we have R t0 = S t0 = 0. Thus the iteration will be terminated in finite steps in the absence of roundoff errors. Next we consider the least Frobenius norm solution of Problem I. Lemma 2.9. [?] Suppose that the consistent system of linear equations Ax = b has a solution x R(A H ), then x is the unique least norm solution of the system of linear equations. Theorem Suppose that Problem I is consistent. If we choose the initial bisymmetric matrix as follows: (2.) A 0 = Y 1 X H + XY1 H + J n (Y 1 X H + XY H Ŝ k 1 )J n + (e p1, e p2,, e ps )Y 2 (e q1, e q2,, e qt ) T + (e q1, e q2,, e qt )Y H 2 (e p 1, e p2,, e ps ) T +J n ((e p1, e p2,, e ps )Y 2 (e q1, e q2,, e qt ) T + (e q1, e q2,, e qt )Y H 2 (e p1, e p2,, e ps ) T )J n, where Y 1, Y 2 are arbitrary n n complex matrices, or more especially, if A 0 = 0, then the solution obtained by Algorithm 1 is the least Frobenius norm solution.
8 1256 Cai Proof. Consider the matrix equations as follows: AX = B, X H A = B H, (e p1, e p2,, e ps ) T A(e q1, e q2,, e qt ) = A S, (e (2.5) q1, e q2,, e qt ) T A(e p1, e p2,, e ps ) = A H S, J n AJ n X = B, X H J n AJ n = B H, (e p1, e p2,, e ps ) T J n AJ n (e q1, e q2,, e qt ) = A S, (e q1, e q2,, e qt ) T J n AJ n (e p1, e p2,, e ps ) = A H S. If A is a solution of Problem I, then it must be a solution of (2.5). Conversely, if (2.5) has a solution A, let Ã = A + AH + J n (A + A H )J n, then it is easy to verify that Ã is a solution of Problem I. Therefore, the consistency of Problem I is equivalent to that of (2.5). By using Kronecker products, (2.5) can be equivalently written as (2.6) X T A I X H (e q1, e q2,, e qt ) T (e p1, e p2,, e ps ) T (e p1, e p2,, e ps ) T (e q1, e q2,, e qt ) T X T J T n J n J T n X H J n (e q1, e q2,, e qt ) T J T n (e p1, e p2,, e ps ) T J n (e p1, e p2,, e ps ) T J T n (e q1, e q2,, e qt ) T J n vec(a) = vec B B H A S A H S B B H Let the initial matrix A 0 be of the form (2.), then by Algorithm 1 and Remark 2.5, we can obtain the solution A of Problem I within finite iteration steps, which can be represented in the same form. Hence we have vec(a ) R X T A I X H (e q1, e q2,, e qt ) T (e p1, e p2,, e ps ) T (e p1, e p2,, e ps ) T (e q1, e q2,, e qt ) T X T J T n J n J T n X H J n (e q1, e q2,, e qt ) T J T n (e p1, e p2,, e ps ) T J n (e p1, e p2,, e ps ) T J T n (e q1, e q2,, e qt ) T J n A S A H S H..
9 Iterative Hermitiangeneralized Hamiltonian solutions 1257 According to Lemma 2.9, vec(a ) is the least norm solution of (2.6), i.e., A is the least norm solution of the (2.5). Since the solution set of Problem I is a subset of that of (2.5), A also is the least norm solution of Problem I. This completes the proof. 3. Iterative algorithm for solving Problem II In this section, we consider iterative algorithm for solving Problem II. For given Ā Cn n and arbitrary A S E, we have A Ā 2 = A Ā+ĀH Ā ĀH 2 2 = A Ā+ĀH +J n(ā+āh )J n 2 + Ā+ĀH J n(ā+āh )J n 2 + Ā ĀH 2 2, which implies that min A Ā is equivalent to A S E min A Ā + ĀH + J n (Ā + ĀH )J n. A S E When Problem I is consistent, for A S E, it follows that { (A Ā+ĀH +J n(ā+āh )J n )X = B (Ā+ĀH +J n(ā+āh )J n)x, (A Ā+ĀH +J n(ā+āh )J n )[p q] = A S ( Ā+ĀH +J n(ā+āh )J n )[p q]. Hence Problem II is equivalent to finding the least norm Hermitiangeneralized Hamiltonian solution of the following problem: ÃX = B, Ã[p q] = ÃS, where Ã = A Ā+ĀH +J n(ā+āh )J n, B = B (Ā+ĀH +J n(ā+āh )J n)x and Ã S = A S ( Ā+ĀH +J n(ā+āh )J n )[p q]. Once the unique least norm solution Ã of the above problem is obtained by applying Algorithm 1 with the initial matrix A 0 having the representation assumed in Theorem 2.10, the unique solution Â of Problem II can then be obtained, where Â = Ã + Ā+ĀH +J n(ā+āh )J n.. A numerical example In this section, we give a numerical example to illustrate the efficiency of the proposed iterative algorithm. All the tests are performed by MATLAB 7. with machine precision around Let zeros(n) denote
10 1258 Cai the n n matrix whose all elements are zero. Because of the influence of the error of calculation, we shall regard a matrix X as zero matrix if X < 1.0e 010. Example.1. Given matrices X and B as follows: X = , B = Let p = (1, 3, 5), q = (1, 2, 6) Q 3,6 and A S = Consider the least Frobenius norm Hermitiangeneralized Hamiltonian solution of the following inverse problem with submatrix constraint: (.1) AX = B and A[1, 3, 5 1, 2, 6] = A S. If we choose the initial matrix A 0 = zeros(6), then by Algorithm 1 and iterating 11 steps, we obtain the least Frobenius norm solution of the problem (.1) as follows: A 11 = with R S 11 2 = e Conclusions In this paper, we construct an iterative method to solve the inverse problem AX = B of the Hermitiangeneralized Hamiltonian matrices with general submatrix constraint. In the solution set of the matrix equations, the optimal approximation solution to a given matrix can.
11 Iterative Hermitiangeneralized Hamiltonian solutions 1259 also be found by this iterative method. The given numerical example show that the proposed iterative method is quite efficient. Acknowledgments Research supported by National Natural Science Foundation of China (No ), Natural Science Foundation of Zhejiang Province (No.Y611003) and The University Natural Science Research key Project of Anhui Province (No. KJ2013A20) References [1] A. BenIsrael and T. N. E. Greville, Generalized Inverse: Theory and Applications, Second Ed., John Wiley & Sons, New York, [2] L. S. Gong, X. Y. Hu and L. Zhang, An inverse problem for Hermitian Hamiltonian matrices with a submatrix constraint, Acta Math. Sci. Ser. A Chin. Ed. 28 (2008), no., [3] N. J. Higham, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl. 103 (1988) [] G. X. Huang and F. Yin, Matrix inverse problem and its optimal approximation problem for Rsymmetric matrices, Appl. Math. Comput. 189 (2007), no. 1, [5] G. X. Huang, F. Yin, H. F. Chen, L. Chen and K. Guo, Matrix inverse problem and its optimal approximation problem for Rskew symmetric matrices, Appl. Math. Comput. 216 (2010), no. 12, [6] M. Jamshidi, An overview on the solutions of the algebra matrix Riccati equation and related problems, Large Scale Systems: Theory and Appl. 1 (1980), no. 3, [7] Z. Jiang and Q. Lu, On optimal approximation of a matrix under a spectral restriction, Math. Numer. Sinica 8 (1986), no. 1, [8] J. F. Li, X. Y. Hu and L. Zhang, Inverse problem for symmetric P symmetric matrices with a submatrix constraint, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no., [9] V. L. Mehrmann, The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution, SpringerVerlag, Berlin, [10] Z. Y. Peng, The inverse eigenvalue problem for Hermitian antireflexive matrices and its approximation, Appl. Math. Comput. 162 (2005), no. 3, [11] Z. Y. Peng, X. Y. Hu and L. Zhang, The inverse problem of centrosymmetric matrices with a submatrix constraint, J. Comput. Math. 22 (200), no., [12] Z. Y. Peng, X. Y. Hu and L. Zhang, The inverse problem of bisymmetric matrices with a submatrix constraint, Numer. Linear Algebra Appl. 11 (200), no. 1,
12 1260 Cai [13] R. Penrose, On best approximation solutions of linear matrix equations, Proc. Cambridge Philos. Soc. 52 (1956) [1] A. J. Pritchard and D. Salamon, The linear quadratic control problem for retarded systems with delays in control and observation, IMA J. Math. Control Information 2 (1985) [15] W. W. Xu and W. Li, The Hermitian reflexive solutions to the matrix inverse problem AX = B, Appl. Math. Comput. 211 (2009), no. 1, [16] Y. X. Yuan, Two classes of best approximation problems of matrices, Math. Numer. Sin. 23 (2001), no., [17] L. J. Zhao, X. Y. Hu and L. Zhang, Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation, Linear Algebra Appl. 28 (2008), no., [18] K. M. Zhou, J. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, New Jersey, [19] F. Z. Zhou, L. Zhang and X. Y. Hu, Leastsquare solutions for inverse problems of centrosymmetric matrices, Comput. Math. Appl. 5 (2003), no , J. Cai School of Science, Huzhou Teachers College, Huzhou, P.R. China
Similarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More information1 Orthogonal projections and the approximation
Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationEXPLICIT ABS SOLUTION OF A CLASS OF LINEAR INEQUALITY SYSTEMS AND LP PROBLEMS. Communicated by Mohammad Asadzadeh. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 30 No. 2 (2004), pp 2138. EXPLICIT ABS SOLUTION OF A CLASS OF LINEAR INEQUALITY SYSTEMS AND LP PROBLEMS H. ESMAEILI, N. MAHDAVIAMIRI AND E. SPEDICATO
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationA note on companion matrices
Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationNotes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 201112 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
More informationSukGeun Hwang and JinWoo Park
Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 471 478 A NOTE ON PARTIAL SIGNSOLVABILITY SukGeun Hwang and JinWoo Park Abstract. In this paper we prove that if Ax = b is a partial signsolvable linear
More informationMarkov Chains, Stochastic Processes, and Advanced Matrix Decomposition
Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More informationLinear Least Squares
Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One
More informationThe MoorePenrose Inverse and Least Squares
University of Puget Sound MATH 420: Advanced Topics in Linear Algebra The MoorePenrose Inverse and Least Squares April 16, 2014 Creative Commons License c 2014 Permission is granted to others to copy,
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationR mp nq. (13.1) a m1 B a mn B
This electronic version is for personal use and may not be duplicated or distributed Chapter 13 Kronecker Products 131 Definition and Examples Definition 131 Let A R m n, B R p q Then the Kronecker product
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationContinuity of the Perron Root
Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSTUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE
STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson AddisonWesley. All rights reserved. Reproduced by Pearson AddisonWesley
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationSign pattern matrices that admit M, N, P or inverse M matrices
Sign pattern matrices that admit M, N, P or inverse M matrices C Mendes Araújo, Juan R Torregrosa CMAT  Centro de Matemática / Dpto de Matemática Aplicada Universidade do Minho / Universidad Politécnica
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationThe Hadamard Product
The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would
More informationUNCOUPLING THE PERRON EIGENVECTOR PROBLEM
UNCOUPLING THE PERRON EIGENVECTOR PROBLEM Carl D Meyer INTRODUCTION Foranonnegative irreducible matrix m m with spectral radius ρ,afundamental problem concerns the determination of the unique normalized
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationExplicit inverses of some tridiagonal matrices
Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25Sep02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationSolution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2
8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More informationVector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationWHEN DOES A CROSS PRODUCT ON R n EXIST?
WHEN DOES A CROSS PRODUCT ON R n EXIST? PETER F. MCLOUGHLIN It is probably safe to say that just about everyone reading this article is familiar with the cross product and the dot product. However, what
More informationADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor
More informationIterative Techniques in Matrix Algebra. Jacobi & GaussSeidel Iterative Techniques II
Iterative Techniques in Matrix Algebra Jacobi & GaussSeidel Iterative Techniques II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin
More informationAN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l4] on graphs and matroids I used definitions equivalent to the following.
More informationInterval PseudoInverse Matrices and Interval Greville Algorithm
Interval PseudoInverse Matrices and Interval Greville Algorithm P V Saraev Lipetsk State Technical University, Lipetsk, Russia psaraev@yandexru Abstract This paper investigates interval pseudoinverse
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationGeneralized Inverse of Matrices and its Applications
Generalized Inverse of Matrices and its Applications C. RADHAKRISHNA RAO, Sc.D., F.N.A., F.R.S. Director, Research and Training School Indian Statistical Institute SUJIT KUMAR MITRA, Ph.D. Professor of
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationBounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices
Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 6 2010 Bounds on the spectral radius of a Hadamard product of nonnegative or positive semidefinite matrices Roger A. Horn rhorn@math.utah.edu
More informationLinear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationMatrix Inverse and Determinants
DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationInstitute of Computer Science. An Iterative Method for Solving Absolute Value Equations
Institute of Computer Science Academy of Sciences of the Czech Republic An Iterative Method for Solving Absolute Value Equations Vahideh Hooshyarbakhsh, Taher Lotfi, Raena Farhadsefat, and Jiri Rohn Technical
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationUsing the Singular Value Decomposition
Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationBASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM
BASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM JORDAN BELL Abstract. This paper gives a basic introduction to the Jordan canonical form and its applications. It looks at the Jordan canonical
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More information26 Ideals and Quotient Rings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationSECRET sharing schemes were introduced by Blakley [5]
206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has
More information7  Linear Transformations
7  Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure
More information