Elliptic Numerical Range of 5 5 Matrices

International Journal of Mathematical Analysis Vol. 11, 017, no. 15, 719-73 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ijma.017.7581 Elliptic Numerical Range of 5 5 Matrices Shyamasree Ghosh Dastidar Department of Mathematics Vidyasagar Evening College Kolkata 700006, India Gour Hari Bera Department of Mathematics St. Paul s C. M. College Kolkata 700009, India Corresponding author Copyright c 017 Shyamasree Ghosh Dastidar and Gour Hari Bera. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we give necessary and sufficient conditions for the numerical range of a 5 5 matrix to be a convex hull of three points and one ellipse or one point and two ellipses. We also find sufficient conditions for which the numerical range of a 5 5 matrix is an elliptic disc. Mathematics Subject Classification: Primary 7A1, Secondary 15A60 Keywords: Numerical range, Associated curve 1. Introduction Let T be a n n complex matrix. The numerical range of T is the subset of complex numbers given by W (T ) = { T x, x : x C n, x = 1}. The following properties of W (T ) are immediate : W (αi + βt ) = α + βw (T ) for α, β C, W (T ) = { λ, λ W (T )},

70 Shyamasree Ghosh Dastidar and Gour Hari Bera W (U T U) = W (T ), for any unitary U. For basic properties of the numerical range a good reference is [,5]. For matrix T, a complete description of the numerical range is well known. It is an elliptic disc with foci at the eigen values λ 1, λ of T and the minor axis of length (tr(t T ) λ 1 λ ) 1 [8]. For a n n complex matrix T, let f T (x, y, z) = det(xh + yk + zi), where H = T +T, K = T T i. Then f T (x, y, z) = 0 defines an algebraic curve of class n with x, y, z viewed as homogeneous line coordinates. The real part of this curve is called associated curve and is denoted by C(T ). The real foci of C(T ) are the eigenvalues of T and W (T ) =convc(t ) [1,7]. For a matrix T of order 3, Kippenhahn [7] studied the factorability of f T in order to describe the numerical range of T. A different procedure is followed in [,3,6] by expressing those conditions in terms of eigenvalues and entries of T, which are easier to apply. We follow the same procedure as in [6] to determine a series of conditions in terms of unitarily equivalent canonical form of a 5 5 complex matrix T for which the numerical range will be an elliptic disc. For a 5 5 arbitrary complex matrix T, we also express those conditions in terms of eigenvalues and entries of T. All these conditions will be useful to construct a 5 5 matrix with an elliptic numerical range.. Main results Let T be a 5 5 complex matrix. It is known that the associated curve C(T ) contains an elliptic disc only when the algebraic curve f T can be factorized as a product of three linear factors and one quadratic factor or one linear factor and two quadratic factors. Here we give necessary and sufficient conditions for which the associated curve C(T ) contains an elliptic disc or discs. By Schur s theorem, every square matrix is unitarily equivalent to an upper triangular matrix. So without loss of generality we can assume that λ 1 a 1 a 13 a 1 a 15 0 λ a 3 a a 5 T = 0 0 λ 3 a 3 a 35 (.1) 0 0 0 λ a 5 0 0 0 0 λ 5 where λ j = α j + iβ j ; α j and β j are real for j = 1,, 3,, 5. Then f T (x, y, z) = det(xre(t ) + yim(t ) + zi 5 ) a α 1 x + β 1 y + z 1 (x iy) a 13 (x iy) a 1 (x iy) a 15 (x iy) a 1 (x + iy) α a x + β y + z 3 (x iy) a (x iy) a 5 (x iy) = a 13 (x + iy) a 3 (x + iy) α a 3x + β 3 y + z 3 (x iy) a 35 (x iy) a 1 (x + iy) a (x + iy) a 3 (x + iy) α a x + β y + z 5 (x iy) a 15 (x + iy) a 5 (x + iy) a 35 (x + iy) a 5 (x + iy) α 5x + β 5 y + z = 5 (α ix + β i y + z) x +y g(x, y, z), where g(x, y, z) is described below. We will use the following notations in writing g(x, y, z) explicitly. By S i1 i i 3 i i 5 we mean the collection of all 5-tuples (i 1, i, i 3, i, i 5 ) of natural numbers such that

Elliptic numerical range of 5 5 matrices 71 1 i 1 < i < i 3 < i < i 5 5 and so on. Then g(x, y, z) = a lm (α i x + β i y + z)(α j x + β j y + z)(α k x + β k y + z) {xre(a kl a lm a km ) + yim(a kl a lm a km )}(α i x + β i y + z)(α j x + β j y + z) S ij,s klm x + y 5 (α i x + β i y + z)p i + (α i x + β i y + z){ x y Re(a jk a kl a lm a jm ) + xy Im(a jk a kl a lm a jm )} + x + y a ij {x Re(a kl a lm a km ) + y Im(a kl a lm a km )} S ij,s klm x + y {x Re(a ij a jk a kl a im a ml ) + y Im(a ij a jk a kl a im a ml )} S ijkl,s iml x + y {x Re(a ij a jk a lm a im a lk ) + y Im(a ij a jk a lm a im a lk )},S im S lk 1 {(x3 3xy ) Re(a 1 a 3 a 3 a 5 a 15 ) + (3x y y 3 ) Im(a 1 a 3 a 3 a 5 a 15 )}, P i = where for each i = 1,,3,,5 S jkm,s lm a jk a lm S jkl,s jml Re(a jk a kl a jm a ml ) S jk,s lkm Re(a jk a lm a jm a lk ), the summations being taken over i j k l m. Henceforth all the summations will be taken over different suffices. This polynomial in x, y, z is denoted by ( ). i.e., 5 f T (x, y, z) = (α i x + β i y + z) x + y g(x, y, z). ( ) At first we prove the following two lemmas which will be used to prove our main result. Lemma.1. Let T be a 5 5 matrix with eigenvalues λ 1, λ, λ 3, λ, λ 5. Then its associated curve C(T ) consists of three points in particular λ 1, λ, λ 3 and one ellipse with foci at λ, λ 5 and the minor axis of length r iff f T (x, y, z) = (α 1 x + β 1 y +z)(α x+β y +z)(α 3 x+β 3 y +z)[(α x+β y +z)(α 5 x+β 5 y +z) r (x +y )], where λ j = α j + i β j, j; α j s and β j s are real. Proof. Along with T, consider the matrix A = diag(λ 1, λ, λ 3 ) ( λ ) r 0 λ 5. Since C(T ) = C(A), the polynomials f T and f A have to be same. Hence the result follows. The converse is clear. Lemma.. Let T be a 5 5 matrix with eigenvalues λ 1, λ, λ 3, λ, λ 5. Then its associated curve C(T ) consists of one point namely λ 1, two ellipses, one with foci at λ, λ 3 and minor axis of length r, the other with foci at λ, λ 5 and minor axis of length s iff f T (x, y, z) = (α 1 x + β 1 y + z)[(α x + β y + z)(α 3 x + β 3 y + z) r (x +

7 Shyamasree Ghosh Dastidar and Gour Hari Bera y )][(α x + β y + z)(α 5 x + β 5 y + z) s (x + y )], where λ j = α j + i β j for all j; α j s and β j s are real. Proof. Along with T, consider the matrix A = diag(λ 1 ) ( λ ) ( r 0 λ 3 λ ) s 0 λ 5. Since C(T ) = C(A), the polynomials f T and f A have to be same. Hence the result follows. The converse is clear. With the help of above lemmas, we now prove the following theorems: Theorem.1. Let T be in upper triangular form (.1). Then its associated curve C(T ) consists of three points and one ellipse if and only if (a) r = S lm a lm, (b) r (λ p + λ q + λ u ) = (c) r (λ p λ q + λ p λ u + λ q λ u ) = (d) r (λ p λ q λ u ) = + a lm (λ i + λ j + λ k ) a kl a lm a km, S klm a lm (λ i λ j + λ i λ k + λ j λ k ) (λ i + λ j )a kl a lm a km + a jk a kl a lm a jm, S ij,s klm S jklm a lm λ i λ j λ k a kl a lm a km λ i λ j S ij,s klm a jk a kl a lm a jm λ i a 1 a 3 a 3 a 5 a 15. If these conditions are satisfied, then C(T ) is the union of three points λ p, λ q, λ u with the ellipse having its foci at two other eigenvalues of T and minor axis of length r. Proof. By Lemma.1, we have f T (x, y, z) = 5 (α i x+β i y+z) r (x +y )(α p x+β p y+z)(α q x+β q y+z)(α u x+β u y+z). Comparing this polynomial with polynomial ( ), we have g(x, y, z) = r (α p x + β p y + z)(α q x + β q y + z)(α u x + β u y + z)

Elliptic numerical range of 5 5 matrices 73 and then obtain the following equalities by comparing the coefficients of x 3, y 3, z 3, x y, xy, x z, xz, y z, yz, xyz respectively. Therefore, (1) r α p α q α u = a lm α i α j α k Re(a kl a lm a km )α i α j S ij,s klm 1 5 P i α i + 1 Re(a jk a kl a lm a jm )α i + 1 Re(a kl a lm a km ) a ij S ij,s klm 1 Re(a ij a jk a kl a im a ml ) 1 Re(a ij a jk a lm a im a lk ) S ijkl,s iml,s im,s lk 1 Re(a 1a 3 a 3 a 5 a 15 ), () r β p β q β u = a lm β i β j β k Im(a kl a lm a km )β i β j S ij,s klm 1 5 P i β i 1 Re(a jk a kl a lm a jm )β i + 1 Im(a kl a lm a km ) a ij S jklm,s i S ij,s klm 1 Im(a ij a jk a kl a im a ml ) 1 Im(a ij a jk a lm a im a lk ) S ijkl,s iml,s im,s lk + 1 Im(a 1a 3 a 3 a 5 a 15 ), (3) r = a lm, S lm () (α p α q β u + α p α u β q + α q α u β p )r = a lm (α i α j β k + α i α k β j + α j α k β i ) Im(a kl a lm a km )α i α j Re(a kl a lm a km )(α i β j + α j β i ) S ij,s klm S ij,s klm 1 5 P i β i + 1 Re(a jk a kl a lm a jm )β i + Im(a jk a kl a lm a jm )α i + 1 a ij Im(a kl a lm a km ) 1 Im(a ij a jk a kl a im a ml ) S ij,s klm S ijkl,s iml 1 Im(a ij a jk a lm a im a lk ) 3 Im(a 1a 3 a 3 a 5 a 15 ),,S im,s lk

7 Shyamasree Ghosh Dastidar and Gour Hari Bera (5) (α p β q β u + α q β p β u + α u β p β q )r = a lm (α i β j β k + α j β i β k + α k β i β j ) Re(a kl a lm a km )β i β j Im(a kl a lm a km )(α i β j + α j β i ) S ij,s klm S ij,s klm 1 5 P i α i 1 Re(a jk a kl a lm a jm )α i + Im(a jk a kl a lm a jm )β i + 1 a ij Re(a kl a lm a km ) 1 Re(a ij a jk a kl a im a ml ) S ij,s klm S ijkl,s iml 1 Re(a ij a jk a lm a im a lk ) + 3 Re(a 1a 3 a 3 a 5 a 15 ),,S im,s lk (6) (α p α q + α p α u + α q α u )r = a lm (α i α j + α i α k + α j α k ) Re(a kl a lm a km )(α i + α j ) 1 5 P i + 1 Re(a jk a kl a lm a jm ), S ij,s klm S jklm (7) (α p + α q + α u )r = a lm (α i + α j + α k ) Re(a kl a lm a km ), S klm (8) (β p β q + β p β u + β q β u )r = a lm (β i β j + β i β k + β j β k ) Im(a kl a lm a km )(β i + β j ) 1 5 P i 1 Re(a jk a kl a lm a jm ), S ij,s klm S jklm (9) (β p + β q + β u )r = a lm (β i + β j + β k ) Im(a kl a lm a km ), S klm (10) α p β q + α q β p + α p β u + α u β p + α q β u + α u β q = a lm (α i β j + α j β i + α i β k + α k β i + α j β k + α k β j ) Re(a kl a lm a km )(β i + β j ) Im(a kl a lm a km )(α i + α j ) S ij,s klm S ij,s klm + Im(a jk a kl a lm a jm ). S jklm Here we see that the combination of (1), (), () and (5) is equivalent to (d), since (1) (5) i()+i() yields (d). Again, the combination of (6), (8), (10) is equivalent to (c) since (6) (8) + i(10) yields (c). Moreover, the combination of (7) and (9) is equivalent to (b) since (7) + i(9) yields (b). This completes the proof. We next prove the following theorem. Theorem.. Let T be in upper-triangular form (.1). Then its associated curve C(T ) consists of one point and two ellipses if and only if

Elliptic numerical range of 5 5 matrices 75 (a) r + s = S lm a lm, (b) r (λ p + λ v + λ w ) + s (λ p + λ q + λ u ) = a lm (λ i + λ j + λ k ) a kl a lm a km, S klm (c) r (λ p λ v + λ p λ w + λ v λ w ) + s (λ p λ q + λ p λ u + λ q λ u ) = a lm (λ i λ j + λ i λ k + λ j λ k ) (λ i + λ j )a kl a lm a km + a jk a kl a lm a jm, S ij,s klm S jklm (d) r {λ p λ v λ w (α p α v α w iβ p β v β w )} + s {λ p λ q λ u (α p α q α u iβ p β q β u )} = a lm λ i λ j λ k a kl a lm a km λ i λ j S ij,s klm + a jk a kl a lm a jm λ i a 1 a 3 a 3 a 5 a 15. If these conditions are satisfied, then C(T ) is the union of one point λ p and two ellipses, one with foci λ q, λ u and minor axis of length r, the other with foci λ v, λ w and minor axis of length s. Proof. By Lemma., we have f T (x, y, z) = (α p x+β p y+z)(α q x+β q y+z)(α u x+β u y+z)(α v x+β v y+z)(α w x+β w y+ z) x +y { r (α p x + β p y + z)(α v x + β v y + z)(α w x + β w y + z) } + s (α p x + β p y + z) (α q x + β q y + z)(α u x + β u y + z) x +y r s (α p x + β p y + z). Comparing this polynomial with polynomial ( ), we have g(x, y, z) = r (α p x+β p y+z)(α v x+β v y+z)(α w x+β w y+z)+s (α p x+β p y+z)(α q x+ β q y + z)(α u x + β u y + z) r s (x + y )(α p x + β p y + z) and obtain the following equalities by comparing the coefficients of x 3, y 3, z 3, x y, xy, x z, xz, y z, yz and xyz respectively. Therefore, (1 ) r α p α v α w + s α p α q α u r s α p = a lm α i α j α k Re(a kl a lm a km )α i α j S ij,s klm 1 5 P i α i + 1 Re(a jk a kl a lm a jm )α i + 1 Re(a kl a lm a km ) a ij S ij,s klm 1 Re(a ij a jk a kl a im a ml ) 1 Re(a ij a jk a lm a im a lk ) S ijkl,s iml,s im,s lk 1 Re(a 1a 3 a 3 a 5 a 15 ),

76 Shyamasree Ghosh Dastidar and Gour Hari Bera ( ) r β p β v β w + s β p β q β u r s β p = a lm β i β j β k Im(a kl a lm a km )β i β j S ij,s klm 1 5 P i β i 1 Re(a jk a kl a lm a jm )β i + 1 Im(a kl a lm a km ) a ij S ij,s klm 1 Im(a ij a jk a kl a im a ml ) 1 Im(a ij a jk a lm a im a lk ) S ijkl,s iml,s im,s lk + 1 Im(a 1a 3 a 3 a 5 a 15 ), (3 ) r + s = a lm, S lm ( ) r {(α v β w + α w β v )α p + (α p β w + α w β p )α v + (α p β v + α v β p )α w } + s {(α q β u + α u β q )α p + (α p β u + α u β p )α q + (α p β q + α q β p )α u } r s β p = a lm (α i α j β k + α i α k β j + α j α k β i ) Im(a kl a lm a km )α i α j Re(a kl a lm a km )(α i β j + α j β i ) S ij,s klm S ij,s klm 1 5 P i β i + 1 Re(a jk a kl a lm a jm )β i + Im(a jk a kl a lm a jm )α i + 1 a ij Im(a kl a lm a km ) 1 Im(a ij a jk a kl a im a ml ) S ij,s klm S ijkl,s iml 1 Im(a ij a jk a lm a im a lk ) 3 Im(a 1a 3 a 3 a 5 a 15 ),,S im,s lk (5 ) r {(α p β v + α v β p )β w + (α v β w + α w β v )β p + (α p β w + α w β p )β v } + s {(α q β u + α u β q )β p + (α p β u + α u β p )β q + (α p β q + α q β p )β u } r s α p = a lm (α i β j β k + α j β i β k + α k β i β j ) Re(a kl a lm a km )β i β j Im(a kl a lm a km )(α i β j + α j β i ) S ij,s klm S ij,s klm 1 5 P i α i 1 Re(a jk a kl a lm a jm )α i + Im(a jk a kl a lm a jm )β i + 1 a ij Re(a kl a lm a km ) 1 Re(a ij a jk a kl a im a ml ) S ij,s klm S ijkl,s iml 1 Re(a ij a jk a lm a im a lk ) + 3 Re(a 1a 3 a 3 a 5 a 15 ),,S im,s lk

Elliptic numerical range of 5 5 matrices 77 (6 ) (α p α v + α p α w + α v α w )r + (α p α q + α p α u + α q α u )s r s = a lm (α i α j + α i α k + α j α k ) Re(a kl a lm a km )(α i + α j ) 1 S ij,s klm 5 P i + 1 Re(a jk a kl a lm a jm ), S jklm (7 ) (α p + α v + α w )r + (α p + α q + α u )s = a lm (α i + α j + α k ) Re(a kl a lm a km ), S klm (8 ) (β p β v + β p β w + β v β w )r + (β p β q + β p β u + β q β u )s r s = a lm (β i β j + β i β k + β j β k ) Im(a kl a lm a km )(β i + β j ) 1 S ij,s klm 5 P i 1 Re(a jk a kl a lm a jm ), S jklm (9 ) (β p + β v + β w )r + (β p + β q + β u )s = a lm (β i + β j + β k ) Im(a kl a lm a km ), S klm (10 ) (α p β v + α v β p + α p β w + α w β p + α v β w + α w β v )r + (α p β q + α q β p + α p β u + α u β p + α q β u + α u β q )s = a lm (α i β j + α j β i + α i β k + α k β i + α j β k + α k β j ) Re(a kl a lm a km )(β i + β j ) Im(a kl a lm a km )(α i + α j ) S ij,s klm S ij,s klm + Im(a jk a kl a lm a jm ). S jklm Here we see that the combination of (1 ), ( ), ( ) and (5 ) is equivalent to (d) since (1 ) (5 ) i( ) + i( ) yields (d). Again, the combination of (6 ), (8 ), (10 ) is equivalent to (c) since (6 ) (8 ) + i(10 ) yields (c). Moreover, the combination of (7 ) and (9 ) is equivalent to (b) since (7 ) + i(9 ) yields (b). This completes the proof. We know that every matrix is unitarily equivalent to an upper-triangular matrix, though it is not easy to obtain the upper triangular form of a matrix of order 5. For generality, we obtain the unitary invariant forms of Theorems.1 and.. Corollary.1. Let T be a 5 5 matrix with eigenvalues λ 1, λ, λ 3, λ, λ 5. Then its associated curve C(T ) consists of three points in particular λ p, λ q, λ u and one

78 Shyamasree Ghosh Dastidar and Gour Hari Bera ellipse having its foci at two other eigenvalues of T and minor axis of length r iff (a) 5 r = tr(t T ) λ i, 5 (b) r (λ p + λ q + λ u ) = r tr(t ) tr(t T ) + λ i λ i, (c) r (λ p λ q + λ p λ u + λ q λ u ) = r λ i λ j S ij (d) + λ i λ i (λ j + λ k + λ l + λ m ) tr(t ) tr(t T ) + tr(t T 3 ), r λ p λ q λ u = r S ijk λ i λ j λ k + S i,s jk λ i λ i λ j λ k tr(t T ) λ i λ j + tr(t ) tr(t T 3 ) tr(t T ). S ij Proof. Let A be in upper-triangular form (.1) which is unitarily equivalent to T. After a little computation, we obtain tr(a A) = tr(a A ) = tr(a A 3 ) = tr(a A ) = 5 λ i + a lm, lm 5 λ i λ i + a ij (λ i + λ j ) + a ij a jk a ik, S ij S ijk 5 λ i λ i + a ij (λ i + λ j + λ i λ j ) S ij + a ij a jk a ik (λ i + λ j + λ k ) + a ij a jk a kl a il, S ijk S ijkl 5 λ i λ 3 i + a ij (λ 3 i + λ i λ j + λ i λ j + λ 3 j) S ij + S ijk a ij a jk a ik (λ i + λ j + λ k + λ iλ j + λ i λ k + λ j λ k ) + S ijkl a ij a jk a kl a il (λ i + λ j + λ k + λ l ) + a 1 a 3 a 3 a 5 a 15.

Using Theorem.1 we have Elliptic numerical range of 5 5 matrices 79 5 (i) r = tr(a A) λ i, 5 (ii) r tr(a) tr(a A ) + λ i λ i = a lm (λ i + λ j + λ k ) a kl a lm a km, S klm (iii) (iv) r S ij λ i λ j + λ i λ i (λ j + λ k + λ l + λ m ) tr(a) tr(a A ) + tr(a A 3 ) = a lm (λ i λ j + λ i λ k + λ j λ k ) (λ i + λ j )a kl a lm a km + a jk a kl a lm a jm, S ij,s klm S jklm r λ i λ j λ k + λ i λ i λ j λ k tr(a A ) λ i λ j S ijk S i,s jk S ij + tr(a) tr(a A 3 ) tr(a A ) = a lm λ i λ j λ k a kl a lm a km λ i λ j S ij,s klm + a jk a kl a lm a jm λ i a 1 a 3 a 3 a 5 a 15. Since trace is unitarily invariant. This completes the proof. Corollary.. Let T be a 5 5 matrix with eigenvalues λ 1, λ, λ 3, λ and λ 5. Then its associated curve C(T ) consists of one point λ p and two ellipses, one with foci λ q, λ u and minor axis of length r, the other with foci λ v, λ w and minor axis of

730 Shyamasree Ghosh Dastidar and Gour Hari Bera length s iff 5 (a) r + s = tr(t T ) λ i, (b) r (λ p + λ v + λ w ) + s (λ p + λ q + λ u ) 5 = r tr(t ) tr(t T ) + λ i λ i, (c) r (λ p λ v + λ p λ w + λ v λ w ) + s (λ p λ q + λ p λ u + λ q λ u ) = r S ij λ i λ j + λ i λ i (λ j + λ k + λ l + λ m ) tr(t ) tr(t T ) + tr(t T 3 ), (d) r {λ p λ v λ w (α p α v α w iβ p β v β w )} + s {λ p λ q λ u (α p α q α u iβ p β q β u )} = r λ i λ j λ k + λ i λ i λ j λ k S ijk S i,s jk tr(t T ) λ i λ j + tr(t ) tr(t T 3 ) tr(t T ). S ij Proof. Proceeding as Corollary.1 and using Theorem. we have 5 (i) r + s = tr(a A) λ i, 5 (ii) r tr(a) tr(a A ) + λ i λ i (iii) (iv) = = = + a lm (λ i + λ j + λ k ) a kl a lm a km, S klm r λ i λ j + λ i λ i (λ j + λ k + λ l + λ m ) tr(a) tr(a A ) + tr(a A 3 ) S ij a lm (λ i λ j + λ i λ k + λ j λ k ) (λ i + λ j )a kl a lm a km + a jk a kl a lm a jm, S ij,s klm S jklm r λ i λ j λ k + λ i λ i λ j λ k tr(a A ) λ i λ j + tr(a) tr(a A 3 ) tr(a A ) S ijk S i,s jk S ij a lm λ i λ j λ k a kl a lm a km λ i λ j S ij,s klm a jk a kl a lm a jm λ i a 1 a 3 a 3 a 5 a 15. Since trace is unitarily invariant. This completes the proof.

Elliptic numerical range of 5 5 matrices 731 Now we are ready to formulate sufficient conditions for a 5 5 matrix T to have an elliptic disc as its numerical range. Theorem.3. Let T be a 5 5 matrix with eigenvalues λ 1, λ, λ 3, λ and λ 5, which satisfies the following conditions : 5 (a) r = tr(t T ) λ i, (b) r (λ p + λ q + λ u ) = r tr(t ) tr(t T ) 5 + λ i λ i, (c) r (λ p λ q + λ p λ u + λ q λ u ) = r S ij λ i λ j (d) + λ i λ i (λ j + λ k + λ l + λ m ) tr(t ) tr(t T ) + tr(t T 3 ), r λ p λ q λ u = r S ijk λ i λ j λ k + S i,s jk λ i λ i λ j λ k tr(t T ) λ i λ j + tr(t ) tr(t T 3 ) tr(t T ), S ij (e) λ λ v + λ λ w λ v λ w + r, where λ = λ p, λ q, λ u and λ v, λ w are other two eigenvalues of T. Then W (T ) is an elliptic disc with foci at λ v, λ w and the minor axis of length r. Proof. By corollary.1, conditions (a) to (d) are equivalent to C(T ) being a union of the three points λ p, λ q, λ u and one ellipse with foci λ v, λ w and minor axis of length r. Moreover condition (e) means that these three points λ p, λ q, λ u lie inside the ellipse. Hence W (T ) is an elliptic disc with foci λ v, λ w and the minor axis of length r. This completes the proof. Theorem.. Let T be a 5 5 matrix with eigenvalues λ 1, λ, λ 3, λ and λ 5. If conditions (a) to (d) of corollary. are satisfied and in addition (e) λ q λ u + λ q λ v + λ u λ w +r λ v λ w +s, (f) λ p λ v + λ p λ w λ v λ w +s. Then W (T ) is an elliptic disc with foci at λ v, λ w and the minor axis of length s. Proof. By corollary., C(T ) consists of one point λ p and two ellipses, one with foci λ q, λ u and the minor axis of length r, the other with foci λ v, λ w and the minor axis of length s. Moreover, for λ in C, such that we have λ λ q + λ λ u λ q λ u + r, λ λ v + λ λ w λ λ q + λ q λ v + λ λ u + λ u λ w λ q λ u + r + λ q λ v + λ u λ w λ v λ w + s

73 Shyamasree Ghosh Dastidar and Gour Hari Bera by condition (e). Then the ellipse with foci λ q, λ u and the minor axis of length r lies inside the ellipse with foci λ v, λ w and the minor axis of length s. Now condition (f) means λ p lies inside the elliptic disc with foci λ v, λ w and the minor axis of length s. This completes the proof. Acknowledgements. We would like to thank Professor Kallol Paul (Jadavpur University) for his invaluable suggestions while preparing this paper. References [1] M. Fiedler, Geometry of the numerical range of matrices, Linear Algebra Appl., 37 (1981), 81-96. https://doi.org/10.1016/00-3795(81)90169-5 [] H.L. Gau and Y.H. Lu, Elliptical Numerical Ranges of Matrices, Master Thesis, National Central Univerty, 003. [3] H.L. Gau, Elliptic Numerical Ranges of Matrices, Taiwanese Journal of Mathematics, 10 (006), 117-18. https://doi.org/10.11650/twjm/150003803 [] K. E. Gustafson and D.K.M. Rao, Numerical Range, Springer-Verlag, Newyork, Inc., 1997. https://doi.org/10.1007/978-1-613-898- [5] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. https://doi.org/10.1017/cbo978051180371 [6] D. S. Keeler, L. Rodman and I. M. Spitkovsky, The Numerical Range of 3 3 Matrices, Linear Algebra Appl., 5 (1997), 115-139. https://doi.org/10.1016/00-3795(95)0067-5 [7] R. Kippenhahn, Über den Wertevorrat einer matrix, Mathematische Nachrichten, 6 (1951), 193-8. https://doi.org/10.100/mana.19510060306 [8] C.K. Li, A simple proof of the elliptical range theorem, Proc. Amer. Math. Soc., 1 (1996), 1985-1986. Received: June 1, 017; Published: August 1, 017