On Joint Eigenvalues of Commuting Matrices. R. Bhatia L. Elsner y. September 28, Abstract

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1 On Joint Eigenvalues of Commuting Matrices R. Bhatia L. Elsner y September 28, 994 Abstract A spectral radius formula for commuting tuples of operators has been proved in recent years. We obtain an analog for all the joint eigenvalues of a commuting tuple of matrices. For a single matrix this reduces to an old result of Yamamoto. Introduction, formulation of the result Let T = (T ; : : :; T s ) be an s-tuple of complex d d-matrices. The joint spectrum pt (T ) is the set of all points = ( ; : : :; s ) 2 C s (called joint eigenvalues) for which there exists a nonzero vector x 2 C d (called joint eigenvector) satisfying T j x = j x for j = ; : : :; s: () If the T 0 i s are commuting then pt(t ) 6= ;. The joint spectrum can be read o the diagonal of the common triangular form: There exists a unitary d d - matrix U such that U H T j U = 0 B@ (j) : : : : : : : : : 0 (j) 2 : : : : : : (j) d CA for j = ; : : :; s: (2) Indian Statistical Institute, Delhi centre, 7, SJS Sansanwal Marg, New Delhi 006, India. Supported by Sonderforschungsbereich 343 "Diskrete Modelle in der Mathematik". y Fakultat fur Mathematik, Universitat Bielefeld, Postfach 003, D-3350 Bielefeld, Germany.

2 Then pt (T ) = f i = ( () i ; : : :; (s) ) : i = ; : : :; dg: We order the joint eigenvalues according to their norms i jj jj : : : jj d jj: (3) Here jj:jj denotes the Euclidean norm in C r and will later on also denote the associated operator norm for matrices. We omit the reference to the dimensions. The s-tuple T can be identied with a linear operator mapping C d into C sd. If S = (S ; : : :; S m ) is another m-tuple of dd-matrices, we dene as T S the sm?tuple of matrices, whose entries are T i S j ; i = ; : : :; s; j = ; : : :m, ordered lexicographically. Continuing in this way we dene T m, consisting of s m entries, each of which is a product of m of the T 0 i s. Identifying again T m with an operator mapping C d into C sm d, T m has d singular values In this note we will prove s (T m ) s 2 (T m ) : : : s d (T m ): (4) Theorem For any s-tuple T = (T ; : : :; T s ) of commuting d d-matrices lim m! (s j (T m )) m = jj j jj j = ; : : :; d: (5) For j = this has been proved in [2], hence we know jj jj = lim m! (s (T m )) m : (6) We also remark that (6) has been proved in [] for l p?norms and in [5] for innite-dimensional Hilbert spaces. If s = then T m is the usual m-th power of T = T, and the joint spectrum is the usual spectrum. For this case (5) has been proved by Yamamoto [6], who showed that for a d d?matrix T with eigenvalues i ordered according to their moduli lim m! (s j (T m )) m = j j j j = ; : : :; d: (7) We will prove Theorem in the following section. 2

3 2 Proof of the Theorem It is convenient to introduce a Kronecker-type matrix product " ~ " in the following way: Let A and B be two (r,s) and (t,u) block matrices A = (A ij ) i=;:::;r; j=;:::;s B = (B ij ) i=;:::;t; j=;:::;u where the A ij and B ij are d d matrices. Dene A ij B = (A ij B kl ) k=;:::;t; l=;:::;u and the rt su - block matrix A ~ B = 0 B@ A B : : : A sb.. A r B : : : A rs B CA (8) of dimension rtd sud. This product is associative. For d = this is the usual Kronecker product, which we will denote by "", following the customary notation (see e.g. [4]). Except for d = however A ~ B is dierent from A B which is an rtd 2 sud 2 matrix. So the product depends on d. However in order to avoid an overload of indices and as we keep d xed throughout, we refrained from stressing this fact in the notation. The main relation for carries over to ~, namely (A ~ B)(C ~ D) = AC ~ BD (9) if all the blocks in B commute with those in C, and the dimensions are tting. For this it suces that AC and BD can be formed. We observe that T m, as dened in the rst section, has the representation T m = T ~ : : : ~ T as the m-fold product of T with itself. First we show that we can transform T to a simpler form without changing the magnitudes involved in (5). Then we prove the Theorem for this simple form using (6) and (7). Let S be a nonsingular d d - matrix, ~T i = ST i S? i = ; : : :; s; 3

4 and ~T = ( ~ T ; : : :; ~ T s ): Obviously the ~ T 0 i s commute too, and pt ( ~ T ) = pt (T ). We show s i ( ~ T m ) jjsjj jjs? jjs i (T m ) i = ; : : :; d; (0) which implies that the lefthand side of (5) is not changed if we replace T m by T ~ m. T m consists of s m blocks of d d matrices C i ; i = ; : : :; s m, each of which is a product of m of the T 0 s. Hence the corresponding block ~ i C i of T ~ m satises Ci ~ = SC i S?. Thus ( T ~ m ) H T ~ m Xs m = ~C H i ~C i () i= Xs m = (S? ) H ( C H i SH SC i )S? (2) i= jjsjj 2 (S? ) H (T m ) H T m S? (3) Here "" is the Loewner partial ordering. Let z 2 C d and x = Sz. The last inequality implies x H ( ~ T m ) H ~ T m x x H x jjsjj 2 jjs? jj 2 zh (T m ) H T m z : (4) z H z Using the Courant-Fischer representation of the eigenvalues : : : d of a hermitean d d matrix B (e.g. [4]) i = min dimv =d+?i max x2v;x6=0 x H Bx x H x for B = ( ~ T m ) H ~ T m and then for B = (T m ) H T m and taking (4) into account, (0) follows. Another transformation of T which doesn't change the numbers jj i jj is the following: Given a unitary s s-matrix U = (u ij ), let W = U I d, where I d is the unit matrix of dimension d, and i.e. ^T i = sx j= ^T = W T; (5) u ij T j i = ; : : :; s 4

5 Then it is obvious that the joint spectrum of ^T is given by the vectors ^i = U i ; i = ; : : :; d, where i 2 pt (T ). Hence jj^ i jj = jj i jj; i = ; : : :; d. Also by using (9) we get ^T m = (W T ) ~ : : : ~ (W T ) (6) = (W ~ : : : ~ W )(T ~ : : : ~ T ) (7) =: W (m) T m : (8) Again by (9) we see that W (m) dened in the last eaquation is a unitary mapping of C sm d into itself, hence s i ( ^T m ) = s i (T m ); i = ; : : :; d: Having now assembled our tools, we invoke a result in ([3], Vol.I, p. 224), by which there P exists a nonsingular d d? matrix S and positive integers t s ; : : :; s t with s i= i = d, such that where ~T i = ~T i = ST i S? = diag( ~ T i ; : : :; ~ T t i ) i = ; : : :; s; 0 B@ ~ i : : : : : :. 0.. : : : CA for i = ; : : :; s = ; : : :; t (9) 0 0 ~ i is an s s? matrix, upper triangular with constant diagonal. Observe that also ( T ~ m ) H T ~ m is block diagonal with s s blocks. This shows that we have to prove (5) only for T 0 i s of the form (9). Clearly then jj jj = : : : = jj d jj. Also by applying a suitable transformation of the form (5), we can assume that T 2 ; : : :; T d have zero diagonals, while the diagonal of T is jj jj. Now from (T m ) H T m (T m ) H T m we get (s (T m )) m (s i (T m )) m (s d (T m )) m i = ; : : :; d: But the leftmost term converges to jj jj by (6), while the rightmost term converges to minj i (T )j = jj jj by (7). Hence (5) holds for i = ; : : :; d. This nishes the proof. 5

6 References [] R. Bhatia and T. Bhattacharyya On the joint spectral radius of commuting matrices Preprint [2] M. Cho and T. Huruya, On the joint spectral radius Proc. Roy. Irish Acad. Sect. A 9, 39-44(99) [3] F.R. Gantmacher,The Theory of Matrices Chelsea, (977) [4] M. Marcus and H. Minc A Survey of Matrix Theory and Matrix inequalities Prindle, Weber and Schmidt, Boston (964) [5] V. Muller and A. Soltysiak Spectral radius formula for commuting Hilbert space operators Studia Mathematica 03(992), [6] T. Yamamoto On the extreme values of the roots of matrices J. Math. Soc. Japan 9, (967) 6