Positive definiteness of tridiagonal matrices via the numerical range

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1 Electronic Journal of Linear Algebra Volume 3 ELA Volume 3 (998) Article Positive definiteness of tridiagonal matrices via the numerical range Mao-Ting Chien mtchien@math.math.scu.edu.tw Michael Neumann neumann@math.uconn.edu Follow this and additional wors at: Recommended Citation Chien, Mao-Ting and Neumann, Michael. (998), "Positive definiteness of tridiagonal matrices via the numerical range", Electronic Journal of Linear Algebra, Volume 3. DOI: This Article is brought to you for free and open access by Wyog Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyog Scholars Repository. For more information, please contact scholcom@uwyo.edu.

2 The Electronic Journal of Linear Algebra. A publication of the International Linear Algebra Society. Volume 3, pp. 93-0, May 998. ISSN ELA POSITIVE DEFINITENESS OF TRIDIAGONAL MATRICES VIA THE NUMERICAL RANGE MAO-TING CHIEN y AND MICHAEL NEUMANN z Dedicated to Hans Schneider on the occasion of his seventieth birthday. Abstract. The authors use a recent characterization of the numerical range to obtain several conditions for a symmetric tridiagonal matrix with positive diagonal entries to be positive denite. Key words. Positive denite, tridiagonal matrices, numerical range, spread AMS subject classications. 5A60, 5A57. Introduction. In a paper on the question of unicity of best spline approximations for functions having a positive second derivative, Barrow, Chui, Smith, and Ward proved the following result. Proposition.. [, Proposition ] Let A =(a ij ) R nn be a tridiagonal matrix with positive diagonal entries. If a ii, a i,i 4 a iia i,i, + +4n i =:::n then det(a) > 0. Johnson, Neumann, and Tsatsomeros improved Proposition. as follows. Proposition.. [6, Proposition.5, Theorem.4] Let A =(a ij ) R be a tridiagonal matrix with positive diagonal entries. If a ii, a i,i < 4 a iia i,i, cos n + i =:::n then det(a) > 0. In addition, if A is (also) symmetric, then A is positive denite. The main purpose of this paper is to derive additional criteria, using the numerical range, for a symmetric tridiagonal matrix A with positive diagonal entries to be positive denite. An example of one of our results is Theorem.. Recall that the numerical range of a matrix A C nn is the set W (A) = fx Ax j x C n jxj =g: Many of our results here will be derived with the aid of the following recent characterization of the numerical range due to Chien. Received by the editors on 3 December 997. Final manuscript accepted on May 998. Handling editor: Daniel Hershowitz. y Department of Mathematics, Soochow University, Taipei, Taiwan 0, Taiwan (mtchien@math.math.scu.edu.tw). Supported in part by the National Science Council of the Republic of China. zdepartment of Mathematics, University of Connecticut, Storrs, Connecticut 0669{3009, USA (neumann@math.uconn.edu). 93

3 94 Mao-Ting Chien and Michael Neumann Theorem.3. [, Theorem ] Let A C nn and suppose S be a nonzero subspace of C n. Then W (A) = [ xy W (A xy ) where x and y vary over all unit vectors in S and S?,respectively, and where x Ax x A xy = Ay y Ax y : Ay Our main results are developed in Section which contains several conditions for a symmetric tridiagonal matrix with positive diagonal entries to be positive denite. In Section 3 we give some examples to illustrate dierent criteria we have obtain in Section. Some notations that we shall employ in this paper are as follows. Firstly, we shall use e () i, i = :::, to denote the {dimensional i{th unit coordinate vector, viz., the i{th entry of e () i is and its remaining entries are 0. Secondly, for a set f :::ng denote by 0 = f :::ngn its complement in f :::ng. Thirdly, if and are subsets of f :::ng and A C nn, then we shall denote by A[ ] the submatrix of A that is detered by the rows of A indexed by and by the columns indexed by. If =, then A[ ] will be abbreviated by A[].. Main Results. We come now to the main thrust of this paper which isto develop several criteria for a symmetric tridiagonal matrix with positive diagonal entries to be positive denite. Suppose rst that A is an n n Hermitian matrix and partition A into R B A = B T where R and T are square. Awell nown result due to Haynsworth [4] (see also [5, Theorem 7.7.6]) is that A is positive denite if and only if R and the Schur complement ofr in A, given by (A=R) =T, B R, B, are positive denite. For tridiagonal matrices, we now give a further proof of this fact based on Theorem.3 and in the spirit of the divide and conquer algorithm due to Sorensen and Tang [9], but see also Gu and Eisenstat [3], in the sense that it is possible to reduce the problem of detering whether A is positive denite into the problem of, rst of all, detering whether two smaller principal submatrices of A are positive denite which can be executed in parallel. Theorem.. Let A C nn be a Hermitian tridiagonal matrix and partition A into A = 6 4 A a + e () 0 a + e ()T a ++ a ++ e (n,,)t 0 a ++ e (n,,) A 3 7 5

4 Positive Deniteness of Tridiagonal Matrices 95 where <<n, A C, and A C n,,n,,. Then A is positive denite if and only if the symmetric tridiagonal matrix () A 0, (=a 0 A ++ ) " ja + j e () e()t a + a ++ e (n,,) e ()T is positive denite. Putting a + a ++ e () e(n,,)t ja ++ j e (n,,) e (n,,)t # () C = A, ja +j e () a ++ e()t a necessary and sucient condition for the matrix in () to be positive denite is that A and A are positive denite and that the inequalities (3), ja +j, A, a ++ > 0 and (4), ja ++ j ja+ a ++ j + (C, a ) ++!,A, > 0 hold. In addition, if A and A are positive denite, then any one of the following three conditions is also sucient for A to be positive denite. (i) The matrices (5) A i, ja +j + ja ++ j a ++ I i i = are positive denite, (ii) (6) (iii) (7),, > ja + j A, a + ja ++j A, ++ > ja+j a ++ d + ja ++j d + where d and d + are the smallest eigenvalues of of A and A,respectively. Proof. Let S be the subspace of C n spanned by e (n) +. Then, by Theorem.3, A is positive denite if and only if A xy is positive denite for all x S and y S?. Assume now that x =[00::: 0 x + 0 ::: 0] t S and y =[y ::: y 0 y + ::: y n ] t S?. Then A xy = a ++ a + x + y + a ++ x + y + a + y x + + a ++ y + x + y Ay

5 96 Mao-Ting Chien and Michael Neumann is positive denite if and only if a ++ y Ay (8), (a + x + y + a ++ x + y + )(a + y x + + a ++ ) > 0: Put y 0 := [y ::: y y + ::: y n ] t C n,. Then (8) becomes a ++ y 0 A 0, y 0 0, ja A + j y y + a + a ++ y y + + a + a ++ y + y + ja ++ j y + y + = a ++ y 0 A 0 y 0 0 A, y 0 " ja + j e () e()t a + a ++ e (n,,) e ()T a + a ++ e () e(n,,)t ja ++ j e (n,,) e (n,,)t # y 0 > 0 This proves the rst part of theorem. Namely, that A is positive denite if and only if the (n, ) (n, ) matrix in () is positive denite. The equivalence of the positive deniteness of the matrix in () to the conditions (3) and (4) is obtained by considering the Schur complement of the matrix C given in () and by utilizing the fact that if u v R`, then (9) det, I + uv T = +v T u: Suppose now that A and A are positive denite. We next establish (i). For this purpose, observe that the subtracted matrix in (), which has ran, has eigenvalues 0 and ja + j + ja ++ j =a ++ so that in the positive semidenite ordering, A 0 0 A, a ++ A 0 0 A ja + j e e T a + a ++ e e T a + a ++ e e T ja ++ j e e T, ja +j + ja ++ j a ++ I n, : Thus if (5) holds, then A is clearly positive denite because then a submatrix of A, namely a,aswell as its Schur complement ina given in () are positive denite. To establish the validity of (ii), factor the rst matrix in () and compute the deterant of the sum of the identity matrix and the ran matrix using the formula in (9). A simple inspection shows that condition (6) implies the positivity of that deterant. Finally, condition (7) implies, condition (6) because, by interlacing, properties of symmetric matrices we have that A, (=d ) > 0 and A, (=d +) > 0. This establishes (iii) and our proof is complete. Next from Gershgorin theorem we now that a sucient condition for an Hermitian matrix A =(a ij ) with positive diagonal entries to be positive denite is that

6 Positive Deniteness of Tridiagonal Matrices 97 P n a ii > ja j=6=i ijj, i = :::n. However, what can be said if, for a tridiagonal matrix, the condition fails to be satised for some rows? In this case, we obtain the following criteria. Theorem.. Let A =(a ij ) C nn be an Hermitian tridiagonal matrix with positive diagonal entries. Then A is positive denite if one of the following conditions is satised. (i) a ii, a ii + ja i+i j p aii ja i,i j + p < a ii : i=35::: aii i=4::: (ii) a ii = a ii, and ja i,i j + ja i+i j : i = 3::: < a ii i=4::: a ii : (iii) a ii 6= a ii, > ja i,i j + ja i+i j, fa ii g, fa ii g ja i,i j + ja i+i j and ja i,i j + ja i+i j : i = 3::: < a ii i=4::: a ii : (iv) a ii 6= a ii, ja i,i j + ja i+i j, fa ii g, fa ii g ja i,i j + ja i+i j and 4 a ii, a ii + ja i,i j + ja i+i j + ja i,i j + ja i+i j 3::: ja i,i j + + ja i+i j, ja i,i j + ja i+i j a ii, a ii < fa iig) ( fa iig i=4::: :

7 98 Mao-Ting Chien and Michael Neumann Proof. We rst note that Bessel's inequality (see, e.g., [7, p.488]) implies that if x and y are n{vectors which are orthogonal to each other, then (0) jx Ayj Ax,jx Axj : Next let S be the subspace generated by e e 3 e 5 :::and observe that vectors x S and y S? have the form x =[x 0 x 3 0 x 5 ] T and y =[0y 0 y 4 ] T, respectively. Then by Theorem.3, A is positive denite if and only if A xy is positive denite for x S and y S?, which is equivalent to () x Ax y Ay > jx Ayj : Assume now that condition (i) is satised. We rst recall the following inequality. Let p p n be positive numbers. Then for any real numbers q q :::q n, () (q + q + + q n )=(p + p + + p n ) fq i =p i : i = :::ng Then by () we have that P jaiixij (3) and P a iijx i j a ii P ja i+ix i + a i+i+ x i+ j (4) = sp s P a iijx i j P ja i+ix i j a + iijx i j ja i+i j a ii + ja i+i j p aii s sp! ja i+i+x i+ j P a iijx i j i=35::: ja i,i j + p : i=35::: aii ja i,i j a ii! From (3), (4), and the hypothesis we now obtain, = Ax,jx Axj x Ax P P ja iix i j P a iijx + ja i+ix i + a i+i+ x i+ j i j P a, iijx i j fa ii g + < i=4::: fa iig i=4::: ja i+i j p aii ja i,i j + p, fa ii g i=35::: aii a ii jy i j = y Ay: a ii jx i j

8 Positive Deniteness of Tridiagonal Matrices 99 The result now follows from () and (0). Let us now set := fa ii g := fa ii g := ja i,i j + ja i+i j, and := ja i,i j + ja i+i j. Then (5) Ax,jx Axj = ja ii x i j + + s ja ii x i j, ja i+i j jx i j + ja ii x i j, ja i+i x i + a i+i+ x i+ j, a ii jx i j s a ii jx i j + ja i+i+ j jx i+ j A 4 = t +(, t), [t +(, t) ] +[t +(, t) ] = t(, t)(, ) +(t +(, t) ) =, (, ) t + (, ) +(, ) t + a ii jx i j (ja i,i j + ja i+i j )jx i j 3 5 for some t [0 ]. If = then the value of (5) is (, )t + : If 6= then (5) assumes its imum 4 (, ) +( + )+(,, ) t = +, (, ) when (, ) (, ) and assumes it imum when (, ) > (, ) : Therefore Ax,jx Axj 8 >< >:,, ( 4, ) +( + )+ if 6= and (, ) (, ) if = or 6= and (, ) > (, ) : (6) If one of the conditions (ii), (iii), or (iv) is satised, then by (6), Ax,jx Axj < ( a ii) ( i=4::: a ii) x Ax y Ay: Thus, by () and (0), A is positive denite. We remar that a similar result can be obtained by switching the odd and even indices in statement of theorem and in the proof. Recall now that for a Hermitian matrix A C nn with eigenvalues ::: n, the spread of A is, n and is denoted by Sp(A). For references on eigenvalue spreads, see, e.g., Nylen and Tam [8], and Thompson [0]. Theorem.3. Let A =(a ij ) C nn be an Hermitian tridiagonal matrix with at

9 00 Mao-Ting Chien and Michael Neumann positive diagonal entries. If fa ii g i=4::: fa ii g > n(n, ) 8, jaii, a jj j +4ja ij j i<j then A is positive denite. Proof. Let S be the subspace generated by e e 3 :::. For x S and y S?, y x = 0. By Remar in Tsing [], W 0 (A) fu Av j u v C n juj = jvj =u v =0g is a circular dis centered at the origin of radius (, n )=. Thus jy Axj 4, n ) = 4 Sp(A).By[8, Corollary 4], (7) But then from (7), Sp(A) Sp(A) n (n, ) n i= Sp(A[fig 0 ]) n n, n, (n, ) (n, 3) (n, 4) ((n, )!)Sp(A[fi jg]) 3 ij from which we now get that 4 Sp(A) n(n, ) 8, jaii, a jj j +4ja ij j : i<j It now follows immediately from () and the fact that x Ax y Ay that A is positive denite. fa ii g i=4::: fa ii g 3. Examples. In this section, we give some examples to illustrate dierent criteria obtained in the paper. Example 3.. Consider the matrix A = 6 4 a 0 0 b b where a and b are positive numbers. Table lists the applicability for positive deniteness of A with a = 4 so that a ii = a ii, while Table shows criteria for a =3:9 and a ii 6= a ii.

10 Positive Deniteness of Tridiagonal Matrices 0 Table Test for positive deniteness of A (Example 3.) Applicability a=4, b=8 a=4, b=7 a=4, b=6. Theorem. (i) yes yes yes Theorem. (ii) yes no no Proposition. yes yes no Gershgorin Theorem no no no Table Test for positive deniteness of A (Example 3.) Applicability a=3.9, b=8 a=3.9, b=7 a=3.9, b=6. Theorem. (i) yes yes yes Theorem. (iii) yes no no Proposition. yes yes no Gershgorin Theorem no no no Example 3.. Consider the matrix A = 4 0 0:5 0 0:5 8 8:5 0 8:5 0 It is easy to verify the principal ors of A are positive, A is positive denite. Table 3 compares three other criteria, and shows that the condition Theorem.3 wors. 3 5 : Table 3 Test for positive deniteness of A (Example 3.) Applicability Theorem.3 Proposition. Gershgorin Theorem yes no no REFERENCES [] David L. Barrow, Charles K. Chui, Philip W. Smith, and Joseph D. Ward. Unicity of best mean approximation by second order splines with variable notes. Mathematics of Computation, 3:3{43, 978. [] Mao-Ting Chien. On the numerical range of tridiagonal operators. Linear Algebra and its Applications, 46:03{4, 996. [3] Ming Gu and Stanley C. Eisenstat. A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. SIAM Journal on Matrix Analysis and Applications, 6:7{9, 995. [4] Emilie V. Haynsworth. Deteration of the inertia of a partitioned Hermitian matrix. Linear Algebra and its Application, :73{8, 968. [5] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, New Yor, 985.

11 0 Mao-Ting Chien and Michael Neumann [6] Charles R. Johnson, Michael Neumann, and Michael J. Tsatsomeros. Conditions for the positivity of deterants. Linear and Multilinear Algebra, 40:4{48, 996. [7] Ben Noble and James W.Daniel. Applied Linear Algebra. Prentice{Hall, Englewood Clis, New Jersey, 988. [8] Peter Nylen and Tin-Yau Tam. On the spread of a Hermitian matrix and a conjecture of Thompson. Linear and Multilinear Algebra, 37:3{, 994. [9] Danny C. Sorensen and Pin-Ta-Peter Tang. On the orthogonality of eigenvectors computed by divide{and{conquer techniques. SIAM Journal on Numerical Analysis, 8:75{775, 99. [0] Robert C. Thompson. The eigenvalue spreads of a Hermitian matrix and its principal submatrices. Linear and Multilinear Algebra, 3:37:333, 99. [] Nam-Kiu, Tsing. The constrained bilinear form and the C-numerical range. Linear Algebra and its Applications, 56:95-06, 984.