Hall, C.A

Porsching, T.A

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 23, 209-212 (1968) 
A Separation Theorem for Nonsymmetric Matrices 
C. A. HALL AND T. A. PORSCHING* 
Bettis Atomic Power Laboratory, 
Westinghouse Electric Corporation, West Miflin, Pennsylvania 15122 
Submitted by G. Birkhoff 
In this paper a separation theorem for the eigenvalues of a symmetric 
matrix is partially generalized to a class of nonnegative matrices. Examples 
are given to show that the hypotheses are not redundant. 
Regarding symmetric matrices, the following result on eigenvalue separa- 
tion is well known [I, p. 115; 2, p. 1031: Let A be an n by n symmetric matrix 
with eigenvalues &[A] satisfying &[A] 3 &[A] > *a* > &[A]. If Ai, the 
principal minor obtained from A by deleting its ith row and column, has eigen- 
dues &[A;] > a-- > h,,[Ai], then 
WI 2 W,I 3 4+&C i = 1, 2,..., n; j = 1, 2 ,..., n - 1. (1) 
The hypothesis of symmetry cannot be omitted (see Example 1 below). 
However, if we restrict our attention to the class of nonnegative matrices, 
then we are able to drop the symmetry condition and still obtain a separation 
theorem. Recall [3, Theorem 2.71 that a nonnegative matrix A always has a 
real eigenvalue h,[A] equal to its spectral radius. With this in mind, we state 
the following 
THEOREM.** Let A be a nonnegative matrix with eigenvalues hi[A], 
i = l,..., n. Let Ai, the ith principal (n - 1) by (n - 1) minor of A, have 
eigenvaZues &[A,], j = l,..., n - 1. If &[A] is any real eigenvalue of A different 
from &[A], then 
WU > Wil 3 4,[4, i = l,..., n, (2) 
with strict inequality on the left if A is irreducible. 
Before proving this theorem, we consider some examples. The first example 
shows that the eigenvalues of a nonsymmetric matrix may not satisfy (1). 
* The Laboratory is operated for the United States Atomic Energy Commission. 
** Professor H. Mint has informed the authors that this theorem is contained in 
the work of G. Frobenius [4], K. Fan and A. Householder [5], and D. Koteljanskii [6]. 
We give a new proof of the theorem. 
209 
210 HALL AND PORSCHING 
EXAMPLE 1. Let A be the companion matrix of (X - 1) (9 - x: + $). 
Then 
UAI = 1, &[A] = 4 + ; i = &[A]. 
But 
h,[A,] = 1 + i/2/2 = h,[A,]; U&J = 2, &[A,1 = 0; 
&[A,] = &[A,] = 0. 
Thus, the eigenvalues of the minors do not separate the eigenvalues of the 
matrix. 
If we allow &[A] to be complex in the above theorem, then we might 
conjecture that 
or possibly 
Re &[A] > Re /\,[AJ > Re &[A], (3) 
I U4 I 3 I U41 I 3 I UAI I , i = 1, 2 )..., n. (4 
Unfortunately, the next example shows that such conjectures are false. 
EXAMPLE 2. Let A be the nonnegative cyclic companion matrix of 
~6 - 1. Then /\,[A] = 1, hj[A] = eitn(i-l)/sl, i = 2,..., 6. But Xi[Ak] = 0, 
k = l,..., 6;j = l,..., 5. Although (2) is satisfied, (3) and (4) are not. 
If all of the eigenvalue of A are assumed to be real in the theorem, we 
cannot replace (2) by the stronger result (1). This is shown by the final 
example. 
EXAMPLE 3. Let A be the nonnegative companion matrix of (X - 1) 
(X + +) (X + t). Then X,[A] = 1, &[A] = - t, &[A] = - 4. But 
W41 = B + & fl, &[A,] = & - 4 16; 
U%,l = t, U41 = 0; &[A31 = h,[A,] = 0. 
For each i, the eigenvalues of Ai do not separate the eigenvalues of A in the 
sense of (1). However, (2) is satisfied. 
PROOF OF THE THEOREM. Clearly, if we prove the result for A,, by 
permutations of rows and corresponding columns of A, the general result will 
follow. 
It is known [3, Theorem 2.71 that &[A] > &[A,]. Moreover, the inequality 
is strict if A is irreducible. Now assume that /\,[A] > h,[A,]. Then the matri- 
ces 
Cj = &[A] I - A, , i = l,p, 
are M-matrices [3, Theorem 3.81 and thus have nonnegative inverses. 
A SEPARATION THEOREM FOR NONSYMMETRIC MATRICES 211 
Consider the matrix 
where 
Let 
xt = - h, ,*a*, %a) and y = - (U2l ,...) a,#. 
and 
Tj = @,[A] I _ A) pj = (Aj[A$- ‘11) “I”;’ 1 . 
Now row reduce Tj to the form 
[ 
(UAl - %I - pc,-zy) 0 
Y 1 I . (5) 
Since the /&[A] are eigenvalues of A for j = 1, p, the rank of Tj must be less 
than n and thus from (5) 
Therefore, 
hj[A] - U11 - J7"Cr'y = 0, (j = 1, p). 
U 11 = h,[A] - gcpy = &)[A] - gC,-l_Y. 
But C, 3 C, . Since these are M-matrices, we have [3, p. 871 that 
This implies that 
Thus, 
/\,[A] - gc,-‘y > X&4] - &V,-‘y > /\,[A] - $“C,-‘_y, 
which contradicts (6). Therefore, 
WJI 2 a%1 2 u4 
(6) 
212 HALL AND PORSCHING 
REFERENCES 
1. R. BELLMAN. “Introduction to Matrix Analysis.” McGraw-Hill, New York, 1960. 
2. J. H. WILKINSON. “The Algebraic Eigenvalue Problem.” Clarendon Press, Oxford, 
1965. 
3. R. S. VARGA. “Matrix Iterative Analysis.” Prentice Hall, Englewood Cliffs, New 
Jersey, 1962. 
4. G. FROBENIIJS. “Uber Matrizen aus positiven Elementen.” S.-B. Kgl. Preztss. 
Akad. Wiss. (1908), p. 471-476. 
5. K. FAN AND A. HOUSEHOLDER. “A Note Concerning Positive Matrices and 
M-Matrices.” Monatshefte f. Ii/lath. 62 (1958), p. 263-270. 
6. D. KOTELJANSKII. “On Some Properties of Matrices with Positive Elements.” 
A.M.S. Translation Series 2, 27 (1963), p. 11. 


A separation theorem for nonsymmetric matrices 

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A separation theorem for nonsymmetric matrices

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