(2 < m < n) submatrix B of A is also normal. By symmetrically permuting rows and columns of A, we can move B to the left upper corner of the matrix A. If we retain ...
Journal of Mathematical Sciences, Vol. 89, No. 6, t998
ON NORMAL
MATRICES WITH NORMAL PRINCIPAL
K h . D. I k r a m o v a n d L. E i s n e r
SUBMATRICES UDC 519.612
Let A be a normal n • n matrix. This paper discusses in detail under what conditions and in what way A can be dilated to a normal matrix of order n + 1 or n + 2. Bibliography: 4 titles.
1. INTRODUCTION We dedicate this paper to Vera Nikolaevna Kublanovskaya on the occasion of her 75th birthday (which, we believe, is only an intermediate jubilee). Since Vera Nikolaevna has never shown m u c h interest in normal matrices, perhaps, we should explain the choice of this topic. In order to justify it, we could mention, first, that normal matrices have always been very loyal with respect to the novelties introduced into numerical linear algebra by Kublanovskaya. For example, they constitute the most broad class of matrices with the property that the basic Francis-Kublanovskaya Q R algorithm converges unconditionally, i.e., with no assumptions on the matrix spectrum (see, e.g., [1, Chap. 4]). Second, t h o u g h indifferent to normality, Vera Nikolaevna has a liking for the consonant notion of normalization (recall the famous normalized decomposition!). Finally, the second author wishes to add that in the turmoil which Russia is now going through, the word "normal" itself is very attractive. Now we state the problem to be investigated in this paper and outline the results. Let A be a normal n x n matrix with real or complex coefficients. Assume that a principal m x m (2 < m < n) submatrix B of A is also normal. By symmetrically p e r m u t i n g rows and columns of A, we can move B to the left upper corner of the matrix A. If we retain the original notation, then A can be represented as follows: A=
D*
"
(1.1)
T h e normality conditions AA* = A*A,
BB* = B*B
imply the relations CC* = DD*, BD + CE* = B*C + DE, D*D + EE* = C*C + E'E,
(1.2) (1.3) (1.4)
which are referred to in what follows as the basic equations. The m a i n problem addressed in this paper is that of determining what information on the blocks C, D, and E in (1.1) can be extracted from the basic equations. However, before tackling this problem for a general n o r m a l matrix A, first we consider the particular case of an irreducible Hessenberg m a t r i x A. This case has been examined recently [2] for matrices resulting from the application of the Arnoldi m e t h o d to an original normal matrix of general type. In [2] it has been shown (in terms of quantities related to the Arnoldi algorithm) that the normality of B implies that B and the w h o l e m a t r i x A are both tridiagonal. In Sec. 2 of this paper, we formulate and prove this assertion as a theorem of pure matrix theory without even mentioning the Arnoldi algorithm. T h e n we return to general normal matrices and consider the above problem for rn = n - 1 and m = n - 2. The case m -= n - 1 is discussed in Sec. 3. In this case, the matrices C and D in (1.1) are vector-columns c and d of dimension r~ - 1, and E is merely a scalar e. It turns out Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 63-94. Original article submitted August 10, 1995. 1072-3374/98/8906-1631 $20.00 9
Plenum Publishing Corporation
1631
that, generically, c and d must be eigenvectors of B associated with the same eigenvalue/3, and the value of e must be specially correlated with/3. In See. 4, we derive a necessary condition for the solvability of the problem considered, which is treated as a dilatation problem. We prove that the columns of the blocks C and D in (1.1) must belong to an invariant subspace of B of dimension, generically, not exceeding four. In the last two sections, we show how to dilate B to a normal matrix A in the case where the columns we wish to a p p e n d to B are taken from a one- or two-dimensional invariant subspace s of B. In the case d i m e = 1, the dilatation problem is always solvable. In particular, this means that the following assertion holds. Theorem
1. Let
all A =
a12 a13 ]
a21
*
*
La31
*
*
(1.5)
be a partial 3 x 3 matrix such that
(1.6)
l a2, 12 + l a31 12=ta12 12 +1a13 12 9 Then a normal completion of A exists.
In the case dim s = 2, the dilatation problem may be unsolvable. In Sec. 5, we provide precise conditions on the columns to be a p p e n d e d that assure the existence of an augmented normal matrix. All the steps of the procedure for constructing a normal completion of B are considered in detail. Some lemmas are only formulated in the paper; their proofs are given in the Appendix. 2. THE HUCKLE PROBLEM
In this section, we prove the following assertion. 2. Let in (1.1) A be an irreducible u p p e r Hessenberg and normal matrix. If a leading principal submatrix B of order rn (2 t'2
for e12 and e21 in terms of f12 and f21. Equating the diagonal entries in (4.2), we arrive at the relations ell
" 3f1~11 = f l t ,
(5.15)
e22 - x2922 = f22.
(5.16)
This shows that although the basic equation (4.3) (or relations (5.9)-(5.10) equivalent to it) implies no explicit constraint on fll and f22, constraints still exist for we want Eqs. (5.15) and (5.16) to be solvable. In accordance with the remark following Lemma 1, the solvability of (5.15) amounts to the relation
(5.17)
fll = (1 -- xl)A1, where A1 is a real number if ~] r 1, and to the relation =- iA1,
fll
A1 E IR,
(5.18)
if ~O = 1. Similarly, for the solvability of (5.16) it is necessary that f22 =
(1-
~2)A2,
A2 E IR.
(5.19)
We do not present a condition similar to (5.18) because here and henceforth we assume that if one of the numbers xl and x2 is equal to 1, then it is Xl, not x2! Assame that conditions (5.17) (or (5.18)) and (5.19) are fulfilled. Then, by Lemma 1, ell = A1 q-7v0,
")' E
e22=A2+6w0,
6CR,
[~,
V0 =
-1"flU0,
w0=x2w--0.
(5.20) (5.21)
Next we show that the values of the free parameters 7 and 5 in (5.20)-(5.21) can be chosen to satisfy the remaining basic equation (4.4). First we examine condition (5.5). Replacing f12 and f21 in (5.13) and (5.14) by their expressions (5.9) and (5.10), we obtain $2f22 + X11`2711
el2
=
1`s(1 - g1~2)
--
e21 = -
'
r2fll + ~2s2f22 rs(1 -~,x2)
(5.22)
(5.23)
Since the denominators of these fractions are complex conjugate numbers, we only need to compare the moduli of their numerators. We have
128 f22+~1r2fll - i~ = 1`4 I f l l is + 84 If22 12 +21`2JRe(-~lf]lf22), i 7"2 fll + ~c2s2f22 I~ =- 1`4 I fll 12 + 84 [ f22 12 q- 21"282Re ( ~ 2 f l l f 2 2 ) , and it remains to verify the equality Re (~-lfllf22)
1642
= Re ( ' ~ 2 f l l f22).
(5.24)
In view of relations (5.17) and (5.19), this amounts to checking whether the product (1- x])(1- ~r2)(~1-~2) is pure imaginary. This fact trivially follows from the identity (1 - ~ q ) ( 1
- 3.r2) (~ 1 - ~2)
= (~1 - 3(1) 4- (~2 - )f2) -t- (3"t'1~2 - "Xl ;'(2).
If fll is given by (5.18) rather than by (5.17), then we must ascertain that the product i ( 1 - x2) (1 - ~2) is pure imaginary, which is pretty obvious. Thus, condition (5.5) is satisfied for arbitrary values of the parameters 7 and/5 in (5.20) and (5.21). Now we consider the off-diagonal entries of the matrices involved in (4.4). We have
{ D * D - Q * D * D Q } = ( 1 - xl~z){D*D}21 = ( 1 - Xl-~2)rs, {-~'E--
EE'}tl
='~12(ell
- e22) - e 2 , ( ~ , i - ~ 2 2 ) .
We must show that there exist 3` and/5 such that the right-hand sides in the above formulas are equal. The most difficult part is to prove that the expression
1
-
e21 1 - ~x~2
a,q~2
( e l l -- e22)
(5.25)
can be represented as a sum of real quantities (and, as a result, is real itself). Introducing the quantities Aa=A1-At, Ba=Tv0-aw0
(5.26)
(see (5.20), (5.21)), we can write
(5.27)
ell -- e22 = A3 + (')'VO --/sWo) -= Aa + B3. If we set 3' =/5 = 0 in (5.27), then (5.25) becomes Aa 1-
-e21) 341~ 2 (~12
n3 ---- r.S I 1 ---X1~2 12 (rtf]l
I1 - ,q Aa
212
4- x2s2f22 - s2722 -- ~ 1 r 2 f 2 2 )
f, 1(1
rs t 1 - x,-~2 I ~ ( A ' r 2
-
I1 - ~
)-
(5.28)
2L2(1 -
).
12 -A2"s2 I1 - ~21
Here, relations (5.22)-(5.23) and (5.17), (5.19) have been used. Expression (5.28) is obviously real. Now we consider the expression eltB3 - e21B3 (5.29) S= 1 - ~1 ~2 We have ~'12/~3 -- e21B3 -~- C12 (3`V0 -- (51-/)0) -- ~21 ("/DO -- /SW0) = ~-12(3`v0 -- /sw0) -- ~21 ( ~ 1 l ) 0
= ~fltvo +
6~2f21w0 =
-- (5~21/)0) ~- "/(~12 -- "~1e21)~)0 -- /5(~-12 -- ~2c21)W0
7f12vo +/sf21wo = --
+ 5 fal~o
(5.30)
1643
In these calculations we have consecutively used relations (5.20)-(5.21), (5.11)-(5.12), and (5.9)-(5.10). In order to show that (5.29) is real, we only need to prove that the numbers ( 1 - ~2 )v0 1- ~
za -
(5.31)
and (1 - ~1 )W0
z; -
1 - g1~2
(5.32)
are real. This is established as follows:
21 --
(1 - ~2)vo
(1 - ~2)~1Vo
(1 - ~2)~'2~a Vo
1 -- Xl'~" 2
1 -- 9a'1~2
~'lM'2 -- 1
~-2 =
(1-.~,)No
1 - ~'1~2
= (1-
~2)~1x2~0
=
g1~2 - 1
(1 - ~2)Vo 9't'a'~2('~l'X2
(1-
~l)w0
>q~2(~1~'2 - t)
_
(1 - ~2)vo
-- 1)
(1-
1 - - ;V~'a~ 2
-- ZI'
~l)W0 _ z2.
1 - gx~z
W h e n introducing expression (5.32) for z2, we have used relation (5.17) for relation (5.18) is used, then we m u s t ascertain that the n u m b e r
fll.
If, in place of (5.17),
i ~o Z'2--
-
-
1 - ~2
is real. Indeed, 7.
Z2 .
-ix2~o
. x2
i Wo
. -
1
.
1
"Z2. -
~2
Thus, we have proved that expression (5.25) is real as the s u m of real quantities (5.28) and (5.29). T h e first term of this s u m is a constant; however, S in (5.29) is a linear function of the free parameters 7 and s Since zl and z2 (or ~2) are obviously nonzero, S depends nontrivially on b o t h 3' and 6, provided t h a t the values of A1 and A2 in (5.17)-(5.19) have been chosen nonzero (which is possible). Now it is clear t h a t by choosing appropriate values for 3' and 6, we can insure that the off-diagonal entries of b o t h matrices in (4.4) are equal. Moreover, this can be achieved in infinitely m a n y ways. Each selection of the values for A1, Az, 7, and ~ completely determines the matrix E and, therefore, the whole m a t r i x A, i.e., a normal extension of the normal matrix B. Our analysis of the case rank D = rank C = 1 is now completed. In all of the subcases, we have established the solvability of the dilatation problem, which can be regarded as a proof of T h e o r e m 1 formulated in the introduction. 6. THE DILATATION PROBLEM FOR m = n -- 2. PART 3 In this section, we continue the analysis of the dilatation problem for m = n - 2. Now, in Stage 2 of the procedure described at the beginning of Sec. 5, we take a pair of eigenvalues of the m a t r i x B. We denote these eigenvalues by /31 and /32. In Stage 3, we specify the columns of D by choosing a pair of linearly independent vectors from the two-dimensional invariant subspace of B associated with/31 and/32. For the matrix A in canonical form, our assumptions imply that only the first two rows of the blocks D and C are nonzero (and linearly independent). Without loss of generality, we can again discard the middle diagonal part of A. This yields a 4 x 4 matrix of the form (1.1) with 2 x 2 blocks B, C, D, and E. There is another possibility of simplifying the problem. On the complex plane, distinct complex numbers /31 and/32 define a unique straight line /3 = ~t + C, t E R , where t is a real parameter, c~ and ~ are some complex constants, and Io~I = 1. Subtracting the scalar matrix ~I from A and multiplying the difference by K, we obtain a new normal matrix with real diagonal block B, B = diag {/31, f12 }, 3,,/32 E R (6.1) 1644
(we use the same nora';ion for the new matrix and its blocks). Observe that the new matrix Q is still diagonal; its diagonal entries are denoted by ~1 and ~2 as previously. For the real diagonal matrix B, the second basic equation (4.3) simplifies to the following form: B D ( I - Q) = D F .
(6.2)
d = det D r 0.
(6.3)
By assumption, D is nonsingular, i.e., Therefore, we can transform (6.2) to the equivalent form D-'BD(I
- Q) = F.
(6.4)
As in Sec. 5, we start by analyzing the case xl = ~2 - x. However, now we must distinguish between the two subcases: x = l and z r If x = 1, then Q = I, and thus (6.4) yields F = 0. In accordance with (4.2), the block E is Hermitian. Moreover, since C = D, any Hermitian 2 x 2 matrix E satisfies the third basic equation (4.4). We conclude that., for a Hermitian B, the requirement C = D insures that a normal extension of B always exists and is itself Hermitian. Consider the case *r ~ 1. Equation (6.4) shows that F must be similar to the diagonal matrix (1 - z ) B , i.e.,
F=
D-1 ((1 - ~ ) B ) D .
D-1B(I-Q)D=
(6.5)
Observe that the rows of D are the left eigenveetors of F corresponding to the eigenvalues ~1(1 - *r) and /32(1 - x), respectively. These numbers are distinct because/31 r Since C = *rD, we have D * D = C ' C , and thus the third basic equation (4.4) means that E must be normal. T h e n F = E - ,rE* must be normal as well, which implies that the rows of D must be orthogonal as eigenvectors of the normal matrix D associated with different eigenvalues. This is a necessary condition imposed on D. Now we prove that this condition is also sufficient, i.e., for a given n u m b e r x r 1 and any nonsingular 2 x 2 matrix D with orthogonal rows, it is possible to dilate B to a normal m a t r i x
A=
E
"
Let D = SW,
S = diag{sl,s2},
be the singular-value decomposition of D. with
sl,s2 > 0,
W*W
= I2,
Performing the unitary similarity transformation (4.8)-(4.9) V = I,~-2 | W * ,
we obtain a new matrix of the form (6.6) with a diagonal matrix D. Since the new matrix F is also diagonar (see (6.5)), we have f l l = /31(1 - - * r ) , f 2 2 = /72(1 - *r). Now we must find a normal 2 x 2 matrix E such that ell
-- X~ll
= /31(1 -- x'),
e22 - - x ' e 2 2 = /32(1 -
x),
(6.7) (6.8)
e l 2 = Je'e-21.
T h e last relation implies the equality
t 12 I=l
I. 1645
It remains to show that by choosing eli, e22, and
e21
appropriately, we are able to satisfy the condition
e - 1 2 ( e l l - e22) = e 2 1 ( ~ l l - - ~ 2 2 ) "
(6.9)
One solution is obvious: set e2~ = 0 and equate e~ and e22 to arbitrary roots of Eqs. (6.7) and (6.8), respectively. It turns out that this is a unique solution. Indeed, using the interrelation between e12 and e21, we rewrite (6.9) as 0 = e21(~etl-
~e22-
e l l q-~22) ~- e21[(e22 - ~n'e22) - (ell - r
= e21(/~2 - /~1)(1 - ~ ) .
(6.10)
Thus, we have shown that, for X 1 = ~2 r 1, the dilatation problem is not always solvable. The requirement that D have orthogonal rows is necessary and sufficient for the solvability of the problem in this case. Now we turn to the main case > r l ~ >r2 and assume that both numbers are not equal to one. Using (6.4), one can derive the following explicit expressions for the entries of F in terms of xi's,/3j's, and dij's:
fla= ~'dl'd22-/32d'2d21(1-x1) =~1(1-~1)-'k d
(/~1-/32)d12d21(1-x1), d
f22 = f12dlld22 - ~ldx2d21 (1 - x2) = / 3 2 ( 1 - x2) -t- (/~2 -/3a)d12d2a (1 - g2), d d
/32)d12d22 (1 -- x2), d
f12 = kl
~l)dlld21
~-
d
(1
-
-
(6.11)
(6.12)
(6.13)
X1).
(6.14)
Equality (4.2) implies relations (5.11)-(5.12) and (5.15)-(5.16). The first two relations yield explicit expressions (5.13)-(5.14) for e12 and e21 in terms of f12 and f21 as in Sec. 5. Next we examine relations (5.15)-(5.16), which, in view of (6.11)-(6.12), take the form e l l - - X l e - l l - - f 1 1 = / ~ 1 ( 1 -- ~ 1 ) [ -
(~l-/~2)d12d22 d
(1-xl),
(6.15)
(1 -
(6.16)
)d12e22 -
=
=
2(1 -
)
d
For a known F , these are equations in ell and e22, respectively. Since the equations e -- ~ l e =
(1 -- ~1),
e -
x2g=
(1
- g2)
are obviously solvable (for example, /~1 and t52 are solutions), the complete solvability of (6.15), (6.16) depends, by Lemma 1, on whether the inclusion d12d21 - C R d
(6.17)
holds. This is certainly true whenever d12 = 0
or
d21 = 0 .
Otherwise, relation (6.17) provides a constraint on the choice of D in the dilatation problem. 1646
(6.18)
Observe that each of the relations (6.17) and (6.18) implies that dl I d22
- -
E R.
(6.19)
Assuming that either (6.17) or (6.18) is fulfilled, we can describe all of the solutions of (6.15) and (6.16) by formulas (5.20) and (5.21), respectively. Define A~ and A2 as follows: A1 = /~1 q-
(/32 -/32)d12d22 d '
(6.20)
A2 =/32 +
(/32 -/~1 )d12d22 d
(6.21)
Obviously, E IR,
A1,As
As + A2 =/31 +/32.
Below we demonstrate that the values of the free parameters 7 and 5 in (5.20)-(5.21) can be chosen in such a way that the remaining basic equation (4.4) is also satisfied. It turns out that inclusion (6.17) implies the equality te12 ]-=] e21 [, (6.22) which is one of the two scalar relations equivalent to the matrix equation (4.4). Indeed, by squaring (6.22) and making use of (5.13)-(5.14), we obtain
[ f12-kxl]21 [2 ] f21 +~r
12.
(6.23)
Upon some simplifications, (6.23) takes the form R e [/1
/21
-
(6.24)
= 0.
By virtue of (6.13) and (6.14), we have f12f21 (('~1 -- "~'2) = (ill
-
-
/32)2dlld22d12d21 d2
(1 - g , ) ( 1
- g2)(xl
- x2)-
(6.25)
As shown in Sec. 5, the product (1 - > f l ) ( 1 - x 2 ) ( K 1
- ~2)
is pure imaginary. Taking into account (6.17) (or (6.18)) and (6.19), we conclude that the right-hand side of (6.25) is pure imaginary as well. Hence, relations (6.24) and (6.22) are valid. It remains to ascertain that the second scalar relation from (4.4) for 7 and 5,
{ D ' D - Q*D*DQ}21 = (1 =
{E'E-
- ~l~2){D*D}21
=
(1 - x1~2) (dlldl2 q- d21d22)
EE*}21 -~-~-12(511--C22 ) --C21 (~-11- ~22),
(6.26)
is satisfied for appropriately chosen values of 7 and s Using (5.13)-(5.14), one can rewrite (6.26) as
I 1 -- Xl~2 [ (dlld12 q-d21622) = (712 q-~lf21)(r
-r
- ( f 2 1 "~
--~22)-
(6.27)
Assume that all of the numbers dij are nonzero and set
~ij = arg dij,
i,j = 1 , 2 .
Note that (6.17) implies
~11 ~t_ ~t922 ___--~.912 -1- ~921 (rood 7r).
(6.28) 1647
This shows that the two s u m m a n d s in the scalar product dlldl2
-~ d 2 1 d 2 2
can differ only by a real factor. Indeed, since their arguments are equal (modulo 7r) to 9911 -- 9912 and 9921 - c222, respectively, it is sufficient to apply (6.28). Now we consider the right-hand side of (6.27). In the same way as in See. 5, we use (5.27) with Ba defined in (5.26) a n d with
A3 = A 1 - A2 = (31- t32) (1-l- 2d12j 2~) = (131- ~2) dlld22 4-d12d2]d Again, first we c o m p u t e the right-hand side of (6.27) in the case where 7 = 5 = 0 in (5.27). This yields A3 [(1 - x2)712 - (1 - ~1)f21]
(6.29)
or (see (6.13)-(6.14))
A3(f11_f12)[ll_~212d12d22~ + 11_~2 ill= dlld21]d "
(6.30)
Since argd
(mod 7r),
= 9911 + 9922 = 9912 "~- ~ 2 1
the two s u m m a n d s in (6.30) can differ only by a real factor. Moreover, their a r g u m e n t is equal (modulo ~-) to 9911 - 9912 = 9921 - 9922 ( m o d 7r) and coincides with the a r g u m e n t of b o t h terms on the left-hand side of (6.27). Now we examine the expression that is obtained when e n -e22 on the right-hand side of (6.27) is replaced by B3. As in (5.30), we find that
-~12B3 - e21Ba = 7 f12vo + ~5f21~o. Multiplying this relation by 1 - ~1x2, we see that in the case is equal to
ell
(/~1 -- '~2) [["Y Td12d22 (1 -- "~'2) (1 ~ ~'1 ~2)v0 - 6 ~ ( 1
- - 122 =
B3, the right-hand side of (6.27)
(1--~'1)--x'l
~2)w0] .
(6.31)
Observe that (1 -
~2)(1
- ~lx2)Vo
=
Zl
I1
-
.>tl~21
~,
(1 - Na)(1
- ~19,,(2)w 0
:
Z 2 I 1 - x~2
12 .
T h e n u m b e r s Zl and z2 were defined in See. 5 via (5.31) and (5.32), respectively, and shown to be real. Once again, we see that the two s u m m a n d s in (6.31) have identical (modulo ~r) arguments that are equal to 9911 - - 9912 ---- 9921 - - 9922 (mod yr). To s u m it up, we have established that both parts of (6.27) can be represented as sums of terms with identical (modulo ~r) arguments. The left-hand side of (6.27) is a constant, whereas the right-hand side is a sum of the constant (6.30) and a nontrivial linear function of 7 and 6 given by (6.31) (recall that now all of the entries dij are considered nonzero). By choosing appropriate values for 3' and (5, we can satisfy relation (6.27). Moreover, this can be done in infinitely many ways. Therefore, if all of the entries of D are nonzero and (6.17) holds, then the dilatation problem is solvable and admits infinitely m a n y solutions. Assume now that dll = 0. Since D is nonsingular, we have d12d21 r O. Condition (6.17) is fulfilled automatically because d = -dlzd21. All of the above arguments remain valid, except that no'a/we have 1648
fewer nonzero terms in each of the relations (6.27), (6.30), and (6.31). If, in addition, d22 = 0, then (6.27) is trivially satisfied because b o t h sides vanish. For a nonzero d22, expression (6.31) is a nontrivial linear function of 7- Hence, there exists a (unique) value of 7 such that b o t h sides in (6.27) are equal. Assume that, instead of dll, some other entry dij is zero. By symmetrically p e r m u t i n g rows and columns of A, one can move this zero entry to the position (1, 1 ) in D. These p e r m u t a t i o n s are merely single similarity transformations of the type (4.5) (4.6) or (4.8)-(4.9) with U2 = V2 =
0
'
or two sequential transformations of these types. Consequently, the canonical form of A will be preserved, t h o u g h the order of the eigenvalues fil and fi2 of the block B m a y be changed. In view of the above analysis of the case dl 1 = 0, we come to the following conclusion: the dilatation p r o b l e m is always solvable, provided that one of the dij's is zero (under the assumption that d r 0). It remains to consider the case xl = 1 r x2. By virtue of (6.11) and (6.14), we have f l l ~-~ 0,
(6.32)
f21 ----~0.
T h e latter equality implies (see (5.12)) that e21 ~ "7~2~12,
and thus (6.22) is satisfied. Equation (6.15) takes the form ell ~ell,
i.e., it only requires t h a t ela be real. If every dij is nonzero, t h e n condition (6.17) is still necessary for the solvability of Eq. (6.16). In expression (6.31), only the first t e r m in the brackets is nonzero. Therefore, the right-hand side in (6.27) is a nontrivial linear function of 7, and thus (6.27) can be satisfied by choosing an appropriate value of 7. T h e above analysis remains almost the same if d n or d21 is zero. T h e only difference is t h a t (6.17) is now automatically fulfilled. This observation also applies to the cases where d12 = 0 or d:2 = 0. However, in b o t h of the latter cases, the left-hand side of (6.27) vanishes. Hence, for the dilatation problem to be solvable, the conditions d21=0 if d a 2 = 0 and d11=0 m u s t be satisfied. This result could be anticipated. equality in (6.32), we conclude that
if
d22=0
Indeed, d12 = 0 implies f12 = 0. C o m b i n i n g this with the second C12 z e21 z 0,
i.e., the matrix E is diagonal. Since, in accordance with L e m m a 3, in a n o r m a l extension of B the columns of D must be orthogonal, we have d21 = 0. T h e same argument is applicable to the case d22 = 0. This remark completes the analysis of the dilatation problem for rn = n - 2. APPENDIX In this Appendix, we prove L e m m a s 1 and 2. We also present a generalization of L e m m a 1 to the case of ~irauItaneou~ complex equations that are linear over the field of reals. Proof of L e m m a 1. Let a = al -t- ia2, b = bl + ib2, f = fa + if2,
z = x + iy,
aa, a2, bl, b2, f l , f2, x, y E IR.
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Then (3.8) is equivalent to the following simultaneous real linear equations in x and y:
(al + bl)X + (b2- a2)y = fl, (a2 + b2)x + ( a l - bl)y---- f2. The determinant of this system equals lal 2 -Ibl 2, which implies the first assertion of the lemma. The second assertion is obvious. Now assume that a and b are given by (3.9). Multiplying both sides of (3.8) by cr (see (3.10)), we arrive at the equation f~ Re(xz) = ~o with x=exp(i
~-cpb)
This justifies condition (3.11). Finally, (3.13) merely expresses the usual interrelation between the solutions of an inhomogeneous and the associated homogeneous linear equation. Observe that in the case considered, we have the tinearity over R, not over C! Proof of Lernrna 2. Let C = UcAcV~,
D = VDADVD
be the singular-value decompositions of the matrices C and D. From (1.2) it follows that Ac = AD --= A. Moreover, one can assume that the left unitary factors in both decompositions are identical, i.e., Uc = UD =-- U.
Then we have C = UAV~,
D = U A D V D.
Hence, C = DQ,
where the matrix Q = VDV~ is unitary. This proves Lemma 2. Now, instead of the scalar homogeneous equation az+bg=O,
consider a system of complex equations of the form A~ + B ~ = 0,
(A1)
where A and B are n x n matrices, tt ~
(
Ul,...
,tt n
,
and the vector g is the complex conjugate of u. Clearly, system (A1) always admits the trivial solution u = 0. We show that a necessary and sufficient condition for (A1) to haire nontrivial solutions can be formulated in the following form: det[ A
B] =0.
(A2)
First we assume that u r 0 is a nonzero solution of (A1). Passing in (A1) to conjugate quantities, we derive B u + A ~ = O.
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Thus, the system of linear homogeneous equations
Av + B w = O, By + An, = 0
(A3)
admits a nontrivial solution, namely, v = u, w = g, which implies (A2). The proof of the sufficiency of (A2) is based on the following observations. If (v, w) is a nonzero solution of (A3), then (N, g) is a solution as well (write conjugate relations for both equations (A3)). It follows that (v + N,~ + w) and (v - N, w - g) also are solutions. Thus, if (A3) admits nontrivial solutions, then among them there must exist nonzero solutions either of the form (u, g), or of the form (u, - ~ ) , or both. Upon substitution of u = iA in (A1), we obtain AA - B ~ = 0.
(A4)
This shows that (A1) admits nontrivial solutions if and only if (A4) does. Assume now that (A2) holds; then (A3) must have nonzero solutions. If, among them, a solution of the form (u,~) exists, then, upon substituting it in the first equation (A3), we obtain (A1), which means that (A1) has a nonzero solution u. Otherwise, (A3) must admit a nontrivial solution of the form ( u , - i f ) . Then (A4) has a nonzero solution, and the same holds for (A1). R e m a r k . If n = 1, A = a, B = b, then (A2) takes the form ]a I = Ibl. Translated by Kh. D. Ikramov. REFERENCES 1. Kh. D. Ikramov, The Nonsymmetric Eigenvalue Problem [in Russian], Nauka, Moscow (1991). 2. T. Huckle, "The Arnoldi method for normal matrices," SIAM J. Matrix Anal. Appl., 15,479-489 (1994). 3. S. H. Friedberg and A. J. Insel, "Hyponormal 2x2 matrices are subnormal," Linear Algebra Appl., 175, 31-38 (1992). 4. P. R. Halmos, "Subnormal suboperators and the subdiscrete topology," Int. Set. Numer. Math., 65, 49-65 (1984).
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