An Elementary Approach to the Jordan Form of a Matrix H. Valiaho ...

An Elementary Approach to the Jordan Form of a Matrix
H. Valiaho
The American Mathematical Monthly, Vol. 93, No. 9. (Nov., 1986), pp. 711-714.
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Mon Jan 28 19:24:46 2008

19861 NOTES 711
Problem 2. Are there reasonable criteria to distinguish those q-series in which the coefficients
in absolute value tend to infinity?
Since I can prove none of Conjectures 1-6, I have little to contribute to the solution of these
more general questions. However progress on Conjectures 1-6 might allow advances on these
more general questions.
It is a very easy matter to compute the power series coefficients for functions such as those in
equations (1)-(7). I will be happy to supply upon request a simple BASIC program implementable
on most small home computers.
This work was partially supported by National Science Foundation Grant DMS-8503324.
References
1. G. E. Andrews, On the theorems of Watson and Dragonette for Ramanujan's mock theta functions, her. J.
Math., 88 (1966) 454-490.
2. G. E. Andrews, Enumerative proofs of certain q-identities, Glasgow Math. J., 8 (1967) 33-40.
3. G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2, G.-C.
Rota ed., Addison-Wesley, 1976.
4. G. E. Andrews, Ramanujan's "Lost" notebook IV. Stacks and alternating parity in partitions, Adv. in Math.,
53 (1984) 55-74.
5. G. E. Andrews, Ramanujan's "Lost" notebook V. Euler's partition identity, Adv. in Math. (to appear).
6. L. A. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans. her. Math.
SOC., 72 (1952) 474-500.
7. P. Erdiis, On an elementary proof of some asymptotic formulas in the theory of partitions, Ann. of Math., 43
(1942) 437-450.
8. G. Meinardus, ~ ber Partitionen mit Differenzenbedingungen, Math. Z., 59 (1954) 388-398.
9. L. J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc.
(2), 16 (1916) 315-336.
10. G. N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936)
55-80.
NOTES
EDITED BY SABRA S. ANDERSON, SI~ELDON AXLER, AND 3. ARTHURSEEBACH, JR.
For instructions about submitting Notes for publication in this department see the inside front cover.
AN ELEMENTARY APPROACH TO THE JORDAN FORM OF A MATRIX
H. V~L~AHO
Department of Mathematics, University of Helsinki, Hallituskatu 15, SF-00100 Helsinki 10, Finland
1. Introduction. There are numerous proofs of the existence of the Jordan normal form of a
square complex matrix. In addition to the classical algebraic and geometric approaches (see, e.g.,
Gantmacher [4]), several elementary proofs have been presented, for example Noble [5], Filippov
[I] (see also [q),Galperin and Waksman [3], and Fletcher and Sorensen [2]. Filippov as well as
Galperin and Waksman establish the existence of the Jordan form by induction, applying the
theory of linear mappings. In an inductive step, Filippov reduces by one the order of every
Jordan block associated with an eigenvalue of the matrix, whereas Galperin and Waksman
determine these blocks entirely. Thus the inductive step of Galperin and Waksman is, in general,
equivalent to several consecutive steps of Filippov, associated with the same eigenvalue. Noble's
proof, like Fletcher's and Sorensen's, are given in terms of matrices. With the aid of the Schur

712 H. VALIAHO [November
unitary triangularization [a,they reduce the determination of the Jordan form of a square
complex matrix to the determination of the Jordan forms of nilpotent matrices. The main result
of Fletcher and Sorensen is refining a strict upper triangular matrix to Jordan form by means of a
sequence of similarity transformations. The pivot of Noble's proof is the construction of the
Jordan strings of a general nilpotent matrix. The proofs of the main theorems of Noble and of
Fletcher and Sorensen are constructive.
We shall propose an elementary inductive proof of the existence of the Jordan form. The basic
idea of the induction is the same as in Galperin and Waksman [3]. The construction of Jordan
strings within the inductive step bears some resemblance to Noble's [5] approach. It is, however,
carried out by a more straightforward method. Contrary to [2] and [5], no preliminary unitary
triangularization is needed, although a simplified version of the proof, using this triangularization,
is outlined in a remark. The only nonconstructive step in the proof is the determination of
the eigenvalues of a matrix.
Of the earlier elementary approaches to the Jordan form mentioned above, only Galperin and
Waksman [3] includes a proof of the uniqueness of the Jordan form; Noble [5] establishes this
result for the part of nilpotent matrices. We give the proof of Galperin and Waksman in a more
elementary and shorter form.
If A E Cmxn (A is a complex m X n matrix), let r(A), g(A) and .&"(A)stand for the rank,
the column space, and the null space of A, respectively. The spectrum of a square matrix A is
denoted by o(A),and the similarity of two square matrices A and B is indicated by A = B. The
identity matrix will be denoted by I and the dimension of a vector space S by dim S. The matrix
with Jl(o) = [o] is called a Jordan block. A Jordan matri; J is a block diagonal matrix
J = Jl @ . . . @J,, where the diagonal blocks 4 are Jordan blocks.
2. Results. We shall give an elementary proof of the following theorem:
THEOREM(Jordan form). For any A E CnXn there exists a nonsingular matrix P E CnX"such
that
where 4 = Jn,(A,),Xi E o(A), i = 1; ..,s. The matrix J, termed the Jordan normal form of A, is
unique up to the order of the Jordan blocks.
REMARK1. Denote the columns of P in (1) by x,~, ...,x ,,,, i = 1,. ..,s, successively. Then
(1) is equivalent to
(2) Axil =Xixil; AX,~=X,X~~+X,,~-~, j=2 ,...,n,,
for i = 1,. . . ,s, where the vectors xi, form a basis of Cn. Any such sequence xil,..., xin, is
called a Jordan string.
REMARK2. Denote B, = A - Ail; it follows from (2) that
Because, in addition, B,xil = 0, we have
xil E ./lr(Bi) n ~(B?I-') and xinZE N(B?~)
Proof of the Theorem. (Uniqueness; cf. [3].) Let X E u(A) be of algebraic multiplicity v and let

19861 NOTES 71 3
there exist, associated with A, q, Jordan blocks of order i, i = 1,. ..,n (set q, = 0 for i > n).
Note that for i = 0,1,2,. .., the rank of JL(o) equals k or max(0, k - i) according to whether
a f 0 or a = 0. Now
implying
n+2
r ( ~ ~ ) = (j-i)qJ+(n-v), i=0,1, ...,n+1.
j=i+l
From these equations we obtain uniquely
(3) q, = r(~'-l)- 2r(Bi) + r(~'+l), i = 1,. . .,n.
(Existence.) We proceed by induction on the order of A. In the inductive step, we first
construct a set of vectors forming the Jordan strings associated with an eigenvalue of A and then
extend this set to a basis of 6".
The theorem is obvious for n = 1. We assume it to hold for matrices of order < n. To
establish the existence of the Jordan form of a matrix A E CnX",we choose any eigenvalue X of
A, define B = A -XI, let p be the smallest positive integer for whch r(BP") = r(BP) =: qo
and, motivated by Remark 2, define the subspaces
Then S, and Jlr(Bi-I), respectively, are the range and the kernel of the linear mapping
whence
We deduce that
dim S, = dim Jlr( B') - dim Jlr( Bi-') = r( B'-') - r( B' )
(0) # Sp c Sp-, c ... c S, =Jlr(B).
The leading vectors y,j of the Jordan strings of B, associated with the zero eigenvalue, are
determined as follows: starting from a basis y,,, .. .,y, of Sp, we extend the basis of S,+
sequentially for i = p - 1, p - 2,. .. ,1, to that of S, Ly means of vectors yil,.. . , y,, (if
S, = S,,,, this sequence is empty). Then, by (4), there exist vectors ti, E Jlr(Bi) such that
y.,= B'-' ti,, j=l, ..., q,, i=l, ...,p.
1J
We shall show that the vectors
(5)
together with any basis
~'-lt,,? ~'-'t,~,. ..,~t,,, t,,, j = I,. .., q,, i = I,. . . ,p
(6) z=[zl,...,zq,,l
of 9(Bp) form a basis of Cn. First, the number of these vectors is
To establish the linear independence of the vectors included in (5-6), let

where /3 is a 9,-vector and the aijk are scalars, implying BPf = Bp(Z/3) = 0. The linear mapping
G: ~ (BP)+ .%(BPf1) = ~ (BP),t * Bt,
is surjective. To see this, note that any 1) E 9(BP") can be expressed in the form y = ~ P+lt=
B(BPt) with t E @", whence y = Bu with u = Bpt E .%(Bp). Because G is surjective, it is an
isomorphism, and so is GP. Thus BP(Zj3) = 0 * Z/3 = 0 * /3 = 0. The equation (7) reduces to
To show that the aijk in (8) are equal to zero, we apply induction on (decreasing) k. From
~p-lf, = 0 we deduce easily that aPjP = 0, j = 1,. . . ,qp. Then, assuming aijk = 0 for j =
1,. . .,qi, i = k,. .. ,p, k = h + 1,..., p, we obtain
implying aijh= 0 for j = 1,.. . ,qi, i = h,... ,p. This completes the proof of the linear indepen-
dence of the vectors included in (5-6).
Because G is an isomorphism, we have BZ = ZC, where C (of order q,) is nonsingular. If X
is the matrix consisting of the vectors (5) in the indicated order, then
where Kl is a Jordan matrix with zero diagonal elements, the blocks of which correspond to the
strings (5). So B = C @ K,. According to the induction hypothesis, C has a Jordan form, say K2.
But then
B = K2 @ K, =: K,
and A = B + XI has the Jordan form J := K + XI.
REMARK3. In the proof of the existence of the Jordan form, the Jordan strings associated with
a given eigenvalue of A are constructed in an inductive step. Thus the proof yields a means of
determining the Jordan form and the Jordan strings of a matrix, provided that its eigenvalues are
given (if only the Jordan form is needed, equation (3) can be applied).
RE~RK4. It is not difficult to reduce the determination of the Jordan form of a square
complex matrix to the determination of the Jordan forms of nilpotent matrices (see, e.g., [5]).It
should be noted that the proof of the Theorem is especially simple in the case of a nilpotent
matrix A f 0: X = 0, B = A, p is the index of A, q, = 0, and 2, C and /3 are void. The proof is
complete in one step; no induction is needed.
References
1. A. F. Filippov, A short proof of the theorem on the reduction of a matrix to Jordan form, Moscow Univ.
Math. Bull., 26 (1971) 70-71.
2. R. Fletcher and D. C. Sorensen, An algorithmic derivation of the Jordan canonical form, this MONTHLY, 90
(1983) 12-16.
3. A. Galperin and Z. Waksman, An elementary approach to Jordan theory, this MONTHLY, 87 (1981) 728-732.
4. F. R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York, 1959.
5. B. Noble, Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1969.
6. I. Schur, ~ber die characteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die
Theone der Integralgleichungen, Math. Ann., 66 (1909) 488-510.
7. G. Strang, Linear Algebra and Its Applications, 2nd ed., Academic Press, New York, 1980.