STABLE MATRICES , THE CAYLEY TRANSFORM , AND CONVERGENT MATRICES

The main result is that a square matrix D is convergent (limn→∞Dn=0) if and 
only if it is the Cayley transform CA=(I−A)−1(I

convergent it is the Cayley transform of a stable matrix (theorem 8).
Cayley transforms are introduced by considering the matrix equation AX+XB C. But first a lemma: Lemma 1" Over field F let matrix A be nxn and let x be either indeterminate over F or in F but not a characteristic value of A. Then (2) Proof: Since x is not a characteristic value of A, (xI-A) exists. (1)follows from (3) Before (2) can be derived, the non-singularity of CA, + must be proven.This equation holds: Therefore, CA + I[ 2xlxI-Al" , 0 since x , 0 and Ix I-A[ , 0 (for xI-A is non-singular); hence, CA, + is non-singular.

QED
CA. of (1) is the generalized Cay!ey transform of A. If x is not a characteristic value of A, then CA, is the Cayley.transform of A; it will be denoted C^.Note that the mapping A-*C^i s bijective from the set of matrices having no characteristic value onto those having no characteristic value -1, the inverse transformation being determined by (2).
Theorem 2": Let matrix A be mxm, G and D be mxn, and B be nxn, all with entries in field F.
where x is either indeterminate over F or in F but , ., 0 and a characteristic value of neither A nor B.
Proof: x satisfies the requirements for CA, and Ca" x to exist, according to the lemma, and the dimensions of CA, Ca,x, (xtm-A)", and (xI,-B) a are such that the expression on the right of (4) is well-defined.QED One consequence of the preceding theorem is the celebrated result that every properly orthogonal'" matrix P can be expressed as P (I+K)(I-K), where K is a real skew matrix.To derive it, in the theorem let F real number field, G I, D O, x -1, and B A'.
Then it follows that A+A' 0 PP' I, where P (-I-A)(-I+A) (I+A)1(I-A), the relationship between P and A being determined by ( 1) and (2) of the lemma (cf. the remark on the bijective character of A-,C^).Likewise the Cayley parametrization of unitary matrices follows [Gantmacher, Vol.I; p. 279 (95)].
Over a field F let A be an mxm matrix, X an mxn matrix and B an nxn matrix.Let ZA,B AX + XB.Clearly the mapping ZA,B: X-AX + XB is a linear transformation on the "This theorem generalizes a lemma of Weyl's [Weyh p. 57, lemma (2.10.A)].
"'An orthogonai matrix is proper none of its characteristic values -1.
linear space of mxn matrices.Denote .tA,A.by .tA:.tA(X)AX + XA*, where all matrices arc of the same dimension.
Corolla 3: Let A, B, G, x, and F be as in theorem 2. Then the mapping G-G-C^.xGCaa is linear from the set of all mxn matrices into itself.This mapping is non- singular ,t:A, is non-singular.
Proof: The linearity of the mapping is obvious.,tA,a is non-singular for cvery D there exists a solution of AX+XB D =, for every E there exists a solution of X-CA.xXCaE (thcorem 2 and the non-singularity of xI,-A and xI-B) the mapping G-,G-C^.xGCt. is non-singular.QED In the rest of this article, let F be the field of complex numbers and let all matrices be square.
The inertia of an nxn matrix X is the ordered triple of integers 0r(X), ,(X) 6(X)) In(X), where n(X) is the number of characteristic values of X whose real parts are positive, ,(X), the number whose real parts are negative, and a(X) the number whose real parts are 0.
QED A square matrix is stable all its characteristic values have negative real parts.S denotes the set of all stable nxn matrices, II denotes the set of all positive-definite hermitian matrices and h/denotcs the set of all negative-definite hermitian matrices.
Theorem 5: A S for any G,II there exists G II G-CAGCA* G there exists G1II G-CAGCA* G1 for some G rl.
Proof: In theorcm 2, let B A*, x (for is not characteristic of a stable matrix and CA presupposes that x ,, 1), and D -Y2(I-A)G(I-A*).Then the last term of ( 4) is G, G-C,xGCA* G,, thcn AG+GA* D; since Gt is arbitrary, so is D, for I-A and I-A* are non- T. HAYNES singular, otherwise C^a nd Ca* CA. would not be defined.Since D tO, A S [Taussky].
Second equivalence: Assume A S. Then 3G elI: AG+GA* D for some D and so G-C^GC^* GI; Gleli as above.Conversely, if, for some GtelI, G-C^GC^* G1 for someG eli, thenAG+GA* DandD eh/.Hence, A eS.

QED
Corollary 6: A S -qG eli: I-diag(g g,) eli, where {g} are the roots of xG-C^GC^* 0; furthermore, & is real (i=l n). Proof: Assume A S. By the first equivalence of the preceding theorem 3G eli: G-C^GC^* I. Since both G and C^GC^* are hermitian and G eli, 3R: R is non-singular and R'GR I, R'(C^GC^*)R diag(g g,,) where {g} are the roots of xG-C,,GC^*I 0.
Since G and C^GC^* are hermitian and G eli, 3R: R is non-singular and R'GR I, R'(C^GC^*)R diag(gt g,,) where {g} are the (real) roots of xG-C^GC^*I 0. Then R "t[I-diag(g ,g,)]R q R "RI-R "ldiag(g g,)R 1 G-C^GC^* eli.By the second equiva- lence of the preceding theorem, A S.

QED
Corollarvfl7: A eS "q G eli: g < (i=l ,n) where {g} are the characteristic values of G'C^GC^* .
Proof: In the preceding corollary, G is non-singular since G ell.Hence, {g,}, the roots of xG-C^GC^*l 0, a the characteristic values of GtC^GC^* , for I,G-C^GCĜ I. x-G-C^GC^*l 0. I-diag(gt g,,) eli is equivalent to i-g, > 0 (i-1 n).

QED
The algebraic properties of the Cayley transform previously developed will be applied to prove theorems about convergent matrices.
Proof: D is convergent D* is convergent.
Assume that D is convergent.Then D* is convergent.By Stein's theorem  [Stein,   1952; p. 82, thm.Proof: By the preceding theorem, D is convergent D C^, where A S. The two equivalences follow from this fact and theorem 5. QED The preceding corollary is a theorem of Taussky's [Taussky; p. 7, thm.5], which is itself a strengthening of Stein's theorem.