325
QUARTERLY OF APPLIED MATHEMATICS
Vol. VIII January, 19S1 No. 4
NEW MATRIX TRANSFORMATIONS FOR OBTAINING
CHARACTERISTIC VECTORS*
By WILLIAM FELLER (Princeton University)
AND
GEORGE E. FORSYTHE (National Bureau of Standards, Los Angeles)
1. Summary of methods. Let A be a non-defective (see §2) square matrix of order n,
symmetric or not, for which it is desired to determine some of the characteristic values
v and associated column-vectors X and row-vectors Y. In terms of matrix products
these quantities are defined by the relations
AX = vX, YA = vY. (1)
A class of numerical procedures is based on iteration methods to obtain one character-
istic value X and the associated vectors C, R. Then A is transformed into a matrix A'
and a new iteration is used to obtain a characteristic value v' and characteristic vectors
X', Y' of A', which can then be converted into corresponding quantities v, X, Y for A.
If more values are wanted, one can continue by transforming A' to A", and so on.
Vector iteration schemes for getting one characteristic value of a matrix were described
in 1929 in [15]; these methods are explained and extended in [1, 13, 8, 9, 5].
In the present paper we are not interested in the iteration procedure as such, but
wish to discuss a class of transformations whereby A' is obtained from A. The earliest
of these known to us is "deflation," suggested by Hotelling [6, 7] for symmetric matrices
and extended in Aitken's thorough study [1] to non-symmetric matrices, defective or
not. In [3] and [4, p. 143] Duncan and Collar introduced a different transformation
(see §3) for non-defective matrices; this was restated in [10] and [16]. It has the ad-
vantage that it reduces the order of the matrix, but it destroys the symmetry. In a
relatively inaccessible paper [13] Semendiaev gave a careful exposition of Aitken's
techniques, and extended them to cover the case of multiple characteristic values in
full generality. His transformation is very general; in the simplest case it somewhat
resembles that of Duncan and Collar. Semendiaev expressed his transformation in the
form of a matrix relation A' = UAU"l. Blanch has devised (unpublished) another
modification of the Duncan-Collar reduction in the form UAU'1.
In [14] Tucker published a related transformation yielding a matrix A' of order
n + 1 which is defective with respect to a double characteristic value zero. Although
the coefficients are obtained easily by bordering A, the increased order may be a dis-
advantage. Tucker's method is not directly a special case of our (5).
A distantly related matrix transformation is the "escalator method" of Morris and
Head [12, 11]. It relates the complete set of characteristic values and vectors of A to
*Received May 20, 1950. The preparation of this paper was sponsored (in part) by the Office of
Naval Research.
326 WILLIAM FELLER AND GEORGE E. FORSYTHE [Vol. VIII, No. 4
the complete set for a submatrix of order n — 1. For this reason the escalator method
cannot be compared to the transformations considered in this paper, in which at each
stage one deals with only two characteristic values of A.
In §2 we present in formulas (5) a four-parameter family of transformations from
A to A'. This family is general enough to include deflation and the procedures of Duncan
and Collar, Semendiaev, and Blanch as special cases; see §3. In §4 two subclasses of
the transformation are discussed: order-reducing and symmetry-preserving transforma-
tions. Two new methods which are both order-reducing and symmetry-preserving appear
promising for practical work with symmetric matrices A. Even for non-symmetric
matrices we feel that our family of transformations offers a choice of procedures which
may occasionally prove useful.
2. The general reduction method. The reduction formulas will be proved in all cases
by means of the following lemma, which can be easily verified.
Lemma. Let the matrix A have the characteristic value v with corresponding column-vector
X and row-vector Y. Let U be a non-singular matrix. Then the matrix A' = UAXJ_1 has
the characteristic value v with corresponding vectors
X' = UX, Y' = YU~\ (2)
We assume for simplicity that A is not defective.* Let X be a (known) characteristic
value of A, with a corresponding column-vector C = {cx , • • • , c„} and row-vector
R — (rj , • • • , r„). Let v be some other (unknown) characteristic value of A, with corre-
sponding column X = [Xi , • • • , x„} and row Y = (yl , ■ ■ ■ , y„). Assume C, R, X, Y
to be so normalized that
RC = YX = 1. (3)
The characteristic values X and v are permitted to be equal, provided that X, Y satisfy
the orthogonality conditions
YC = RX = 0, (4)
which are automatic when X v.
Let y, p, 13, t be complex parameters, and put for abbreviation
r = 1 — ycn , P = 1 - prn , f = TP — yp.
We now introduce a new matrix A' = A'(y, p, /3; t), defined for all values of the param-
eters except when /3 = 0 and at the same time f 0:
a'a = au — yCiani — prjain + ypctrj(ann — X) — /fc.r, ,
a'in = — P[ain - yCiann — fc;r„r + (X - t)yCi\,
-/i8_1[a»; - prjann - fc„r-,P + (X - t)prt (if /3 ^ 0)
0 (if / = p = 0) (1 < i < n - 1, 1 < j < n - 1),
aL = f(am - tc„rn) + (X - i)(l — /).
a'- =
(5)
*In the terminology of [9] a defective matrix A is one for which no transform PAP 1 is a diagonal
matrix. An equivalent definition is that A has one or more non-linear elementary divisors; see [2].
1951] MATRIX TRANSFORMATIONS FOR OBTAINING CHARACTERISTIC VECTORS 327
For all values of the parameters a matrix U = U(y, p, (3) will be defined below, with
the property that
A'(y, p, 0; t) = U(A - tCR)U~\ (6)
By the lemma A' has the characteristic values of A — tCR, i.e., those of A except that
the single characteristic value X is changed to X — t. In particular, v is a characteristic
value of A', and the vectors X, Y are transformed into characteristic vectors X' =
TJX and Y' = YU~l of A' corresponding to v. By use of the formulas for U given below,
it is easily shown in each case that
x'i = Xi — yCiXn (1 < i < n — 1),
(if 0 ^ 0)
Xn -
(if / = 0 = 0),
y'i = Hi - pfiVn (1 < j <n - 1), y'n = -Pyn .
(7)
The vectors C, R are transformed into characteristic vectors C' = UC, R' = RLT1 of
A' corresponding to X — t. The formulas for the components of C, R' are given for
each case below, as are the definitions of U, IT"1.
An arbitrary choice of the parameters y, p, 0, t may be used to analyze the matrix
A. Knowing X, C, R, one computes A' from (5). By iteration or otherwise one next
determines a new characteristic value of A'. By the lemma v is also a characteristic
value of A, and the vectors of A corresponding to v may be calculated from the relations
(7); these give xk and yk in terms of x'k and y'k except when / = 0. (The case (3 = 0 is
included in the exceptional case / = 0, as was stated before (5).) When / = 0, one first
gets xn , yn from formulas (8), which are derived from (7), (4), and (3):
n—1
X„ = — (7 + r„r)_1 2 riXi >
t-1
(8)
Vn = -(p + c„P)_1 y'fii .
i = l
If still more characteristic values of A are desired, one can use v, X', Y' in (5) to trans-
form A' into A", and so on. The procedure is useful to the extent that it is easier to
find v, X', Y' from A' than it is to find v, X, Y from A.
It remains only to exhibit U = U(y, p, 0), so that the reader may verify equation (5).
For completeness C and R' are also given. There will be three cases. All matrices are
exhibited in a partitioned form, with a square matrix of (n — l)-th order at the upper
left.
Case 1. r 9^ 0, ^ 0. Here we define
U = U(y, P, 0)
X-
328 WILLIAM FELLER AND GEORGE E. FORSYTHE [Vol. VIII, No. 4
where Sa = 0 (tV j) and 5,,- = 1 (i = j). It can be verified that
/ Sa — ypT or,
U' =
V-pr.T"1
-/sr_,7Ci
-isr-1
It is found that
c'i = Tci(l < i < n — 1), ci = — p/3_1 — fcnj8-1;
r'i = /r~V,(l < j < n - 1), K = -j9rr_1 - /3r„
Case 2. T = 0. /3 5^ 0. Here 7 = <£l, and we define
'Sa + CJi
\—(2p + c„P)/3~V,-
' Si, — c<r, + pc~\r„cn - l)c,r,
E/ = c/fc1, P, js) =
*7-' =
\ — (2p + c„P)r,
It is found that
CiC~ (rncn - 1) N
p/3-c;1 — (2 p + c„P)/TV„y
fic,c~\r„cn — 1) \
—/3 + /3(r„c„ — 1)/
c5 = C;(l < i < n — 1), ci = —(p + c„P)/3 \
r? = —prfin\ 1 < j < n — 1), ri = —/8<£\
Case 3. / = 0 = 0. Here 7, p are restricted to such values that / =
(1 — 7C„)(1 — pr„) — 7p = 0. We define
U = 17(7, P, 0) =
IT--|
It is found that
Sa — Tc>r,
-7Ci — rc;0
5,,- — Pc,r,
v - pTj - Pc„r,
cj = 0 (1 < i < n — 1), ci = 1,
r'i = 0 (1 < j < n - 1), r'n = 1.
3. Known special cases of our transformation. For certain values of the parameters
7, p, j8, t the matrices A' defined by (5) have previously been used to analyze matrices
A. We know of the following special cases:
(a) Duncan and Collar [3] and [4, p. 143]: Case 2, with 7 = ci"1, p = 0, /3 = —1,
t = X. (Rows and columns have been interchanged in our presentation.) The
matrix A' is the result of subtracting from each of the other rows ci"1 times the
matrix product of C by the last row of A.
1951] MATRIX TRANSFORMATIONS FOR OBTAINING CHARACTERISTIC VECTORS 329
(b) Hotelling [6] (deflation): Case 1, with y = p = 0, /3 = — 1, < = X. Here U is
the identity matrix.
(c) Semendiaev [13, p. 212]: Case 3, with y = c"1, p = /3 = t — 0. This is only a
special case of Semendiaev's general reduction.
(d) Blanch (unpublished procedure used at the National Bureau of Standards, Los
Angeles): Case 1, with y = 0, p = r~\ /3 = 1, t = 0. To compare method (d)
with (a), to which it is closely related, one should interchange rows and columns.
4. New special cases of our transformation. One useful class of matrix transforma-
tions consists of those in §2 for which / = 0; here /3 and t remain unrestricted. For these
it is seen that a'ni = 0 (j = 1, 2, • • • , n — 1), so that A' is essentially reduced to order
n — 1. We call these transformations order-reducing; by their use subsequent iterations
become shorter. The methods (a), (c), (d) in §3 are order-reducing.
Another special class of matrix transformations consists of those in §2 for which
y = p, (82 = /, with t unrestricted. When A is symmetric it is reasonable to pick c,- = r, ,
and then A' is also symmetric; hence this class of transformations is called symmetry-
preserving. In §3 only method (b) is symmetry-preserving.
New transformations which are both symmetry-preserving and order-reducing are
those in Case 3 of §2 for which p = y. Except for the unessential freedom allowed t,
there are commonly two of these transformations. When c< = r,- these may be defined by
Y - P = (c„ + 1)_1 = (r„ + I)"1, (9)
y = p = (cn - I)"1 = (r„ - I)"1. (10)
Symmetric matrices are more convenient to deal with than non-symmetric ones, in that
by their use the storage requirement is approximately halved and the round-off errors
are more easily estimated. For dealing with symmetric matrices A, therefore, the
transformations defined by (9) and (10) look promising. For non-symmetric matrices
A the method (d) of §3 seems quite satisfactory, but it may occasionally be useful to
have other subcases of (5) available.
5. Numerical example. From [8, p. 327] we obtain the symmetric matrix*
-2 -2 0 3-1
-2 0-3 5 0
A = 0-3-5 1 1
3 5 1-3-1
,-1 0 1 -1 -1
By an iteration one can obtain the dominant characteristic value X = —9.88649 and
corresponding normalized row-vector
R = (-.35616, -.52348, -.46374, .61437, .08124).
Since A is symmetric, the column-vector C has the same components.
*We have corrected a misprint in [8], Mr. William Paine of the National Bureau of Standards,
Los Angeles, assisted with the calculations.