# MATRICES if AB = I, then BA = I (and conersely)

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### Alexander Abian

Mar 16, 1999, 3:00:00 AM3/16/99
to
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Without invoking the notions of the characteristic polynomial, minimal
polynomial or Cayley-Hamilton Theorem, without any of these I will
prove in the most elementary (but rigorous and emotionally charged)
way that:

For matrices A, B (say over real numbers) if AB = I then BA = I

With absolutely no loss of generality, I will consider that A and B
are square 2 by 2 matrices and that

(1) AB = I

Since A is a 2 by 2 matrix, as such it is a four dimensional vector
and therefore the 5 vectors

A^4 A^3, A^2, A , I

must be linearly dependent, i.e.,

aA^4 + bA^3 + cA^2 + dA + eI = 0
(2)
where a, b, c, d, e are real numbers NOT ALL o (zero)

If e =/= o then from (1) it follows that

- 1/e (aA^3 + bA^2 + cA + dI) A = I

and therefore the matrix K given by

K = - 1/e (aA^3 + bA^2 + cA + dI)

is the two-sided inverse of A , i.e.,

KA = AK = I and thus KAB = AKB = B which by (1) implies

(3) K = B

and since K is the two-sided unique inverse of A we see by (3) that
B is also the unique two sided inverse of A and therefore

(4) AB = BA = I, as desired.

Now, if e in (2) is o then let, say, b be the nonzero coefficient
of the smallest power of A in (2). Then consider

(5) aA^3 + bA^2 = 0 with b =/= o

from which it follows that

(aA + bI)A^2 = 0 and thus (aA + bI) AABB = 0 which by (1) implies

- a/b A = I

and therefore the matrix K given by K = -a/b I is the two sided
inverse of A from this, as in the case of (3), we conclude that in this
case also AB = BA = I, as desired.

REMARK. The proof of general case of n by n matrices readily follows the
pattern of my proof given above.

--
-------------------------------------------------------------------------
ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND EPIDEMICS.
JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR SYSTEM.
REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN EARTH(1990)
THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF SPACE

### David Petry

Mar 17, 1999, 3:00:00 AM3/17/99
to
Here's another simple proof that AB = 1 implies BA = 1
for nXn matrices A and B.

Clearly the theorem holds if A is invertible, hence it holds
generically (on an open dense set of matrices), hence it
holds in the field of symbols, hence it's true. QED

### Robin Chapman

Mar 18, 1999, 3:00:00 AM3/18/99
to

Dense sets? Proving a simple algebraic result by appealing to
topological properties of infinite sets of points? All
a bit Cantorian isn't it?

--
Robin Chapman + "Going to the chemist in
Department of Mathematics, DICS - Australia can be more
Macquarie University + exciting than going to
NSW 2109, Australia - a nightclub in Wales."
rcha...@mpce.mq.edu.au + Howard Jacobson,
http://www.maths.ex.ac.uk/~rjc/rjc.html - In the Land of Oz

### John Rickard

Mar 18, 1999, 3:00:00 AM3/18/99
to
David Petry <david...@mindspring.com> wrote:
: Here's another simple proof that AB = 1 implies BA = 1
: for nXn matrices A and B.
:
: Clearly the theorem holds if A is invertible, hence it holds
: generically (on an open dense set of matrices), hence it
: holds in the field of symbols, hence it's true. QED

I don't know exactly what is meant by "the field of symbols", but I
think there must be something wrong with this proof. For it seems
that the same argument would give:

Theorem: If A and B are nxn matrices with AB = 0, then B = 0.

Proof: Clearly the theorem holds if A is invertible, hence it

holds generically (on an open dense set of matrices), hence
it holds in the field of symbols, hence it's true.

Corollary: 1 = 0. (Take n = 1, A = (0), B = (1).)

--
John Rickard <John.R...@virata.com>

### David Petry

Mar 18, 1999, 3:00:00 AM3/18/99
to

John Rickard wrote in message ...

That's clever.

Let me state the proof more carefully.

Note that for every invertible A, if AB = 1, then AB-BA = 0 and each
element of AB-BA is a polynomial expression in the elements of A.
Since "AB-BA = 0" holds generically, it holds always.

### David Petry

Mar 18, 1999, 3:00:00 AM3/18/99
to

Robin Chapman wrote in message <36F02136...@mpce.mq.edu.au>...

>David Petry wrote:
>>
>> Here's another simple proof that AB = 1 implies BA = 1
>> for nXn matrices A and B.
>>
>> Clearly the theorem holds if A is invertible, hence it holds
>> generically (on an open dense set of matrices), hence it
>> holds in the field of symbols, hence it's true. QED
>
>Dense sets? Proving a simple algebraic result by appealing to
>topological properties of infinite sets of points? All
>a bit Cantorian isn't it?

That's an interesting remark.

I was searching for examples of proofs which use topological
arguments to prove theorems in discrete mathematics. The above
method provides a slick proof of the theorem that a matrix satisfies
its characteristic polynomial. So the method of proof was something
that caught my eye because of my interest in "Cantorian" math.

### David Petry

Mar 18, 1999, 3:00:00 AM3/18/99
to

David Petry wrote in message <7csg54\$1uf\$1...@samsara0.mindspring.com>...

>
>John Rickard wrote in message ...
>>David Petry <david...@mindspring.com> wrote:
>>: Here's another simple proof that AB = 1 implies BA = 1
>>: for nXn matrices A and B.
>>:
>>: Clearly the theorem holds if A is invertible, hence it holds
>>: generically (on an open dense set of matrices), hence it
>>: holds in the field of symbols, hence it's true. QED
>>
>>I don't know exactly what is meant by "the field of symbols", but I
>>think there must be something wrong with this proof. For it seems
>>that the same argument would give:
>>
>> Theorem: If A and B are nxn matrices with AB = 0, then B = 0.
>>
>> Proof: Clearly the theorem holds if A is invertible, hence it

>> holds generically (on an open dense set of matrices), hence
>> it holds in the field of symbols, hence it's true.
>>
>> Corollary: 1 = 0. (Take n = 1, A = (0), B = (1).)
>
>That's clever.
>
>Let me state the proof more carefully.
>
>Note that for every invertible A, if AB = 1, then AB-BA = 0 and each
>element of AB-BA is a polynomial expression in the elements of A.
>Since "AB-BA = 0" holds generically, it holds always.

I wish I could cancel that article. It's totally wrong, as Rickard's
example shows. There are cases where the method of proof
works, but this is not one of those cases, as far as I can tell.

### Robin Chapman

Mar 19, 1999, 3:00:00 AM3/19/99
to
John Rickard wrote:
>
> David Petry <david...@mindspring.com> wrote:
> : Here's another simple proof that AB = 1 implies BA = 1
> : for nXn matrices A and B.
> :
> : Clearly the theorem holds if A is invertible, hence it holds
> : generically (on an open dense set of matrices), hence it
> : holds in the field of symbols, hence it's true. QED
>
> I don't know exactly what is meant by "the field of symbols", but I
> think there must be something wrong with this proof. For it seems
> that the same argument would give:
>
> Theorem: If A and B are nxn matrices with AB = 0, then B = 0.
>
> Proof: Clearly the theorem holds if A is invertible, hence it
> holds generically (on an open dense set of matrices), hence
> it holds in the field of symbols, hence it's true.
>
> Corollary: 1 = 0. (Take n = 1, A = (0), B = (1).)

Look's like David's argument fails this "reality check".

### Rick Decker

Mar 21, 1999, 3:00:00 AM3/21/99
to
David Petry wrote:
>
[...]

>
> I was searching for examples of proofs which use topological
> arguments to prove theorems in discrete mathematics.

[...].

Have you seen the topological proof of the infinitude of primes?
It's also pretty "slick."

Regards,

Rick

-----------------------------------------------------
Rick Decker rde...@hamilton.edu
Department of Comp. Sci. 315-859-4785
Hamilton College
Clinton, NY 13323 = != == (!)
-----------------------------------------------------

### Edward C. Hook

Mar 22, 1999, 3:00:00 AM3/22/99
to
In article <36F17574...@mpce.mq.edu.au>,

Robin Chapman <rcha...@mpce.mq.edu.au> writes:
|> John Rickard wrote:
|> >
|> > David Petry <david...@mindspring.com> wrote:
|> > : Here's another simple proof that AB = 1 implies BA = 1
|> > : for nXn matrices A and B.
|> > :
|> > : Clearly the theorem holds if A is invertible, hence it holds
|> > : generically (on an open dense set of matrices), hence it
|> > : holds in the field of symbols, hence it's true. QED
|> >
|> > I don't know exactly what is meant by "the field of symbols", but I
|> > think there must be something wrong with this proof. For it seems
|> > that the same argument would give:
|> >
|> > Theorem: If A and B are nxn matrices with AB = 0, then B = 0.
|> >
|> > Proof: Clearly the theorem holds if A is invertible, hence it
|> > holds generically (on an open dense set of matrices), hence
|> > it holds in the field of symbols, hence it's true.
|> >
|> > Corollary: 1 = 0. (Take n = 1, A = (0), B = (1).)
|>
|> Look's like David's argument fails this "reality check".
|>

Actually, shouldn't it be remarked that the theorem in question
(the "AB = I ==> BA = I" one) only _applies_ to invertible matrices ??
In which case, David's argument assumes what's supposed to be proved
(in addition to using nuclear weapons to swat a mosquito) ...

Of course, I've noticed that this thread has suffered from a certain
amount of confusion, no doubt engendered by the fact that the result
in question really belongs at the very beginning of any discussion of
finite-dimensional linear algebra, so there's (no doubt) a school of
thought that would say that I'm wrong in my observation.

I think that I'd award the prize for the best proof/discussion so far
to Abian ...

--
Ed Hook | Copula eam, se non posit
MRJ Technology Solutions, Inc. | acceptera jocularum.
NAS, NASA Ames Research Center | I can barely speak for myself, much
Internet: ho...@nas.nasa.gov | less for my employer

### Bill Dubuque

Apr 13, 1999, 3:00:00 AM4/13/99
to
There is a non0 poly P with P(A)=0 (via dim V < oo => dim Hom(V,V) < oo)
We may assume P(0) non0 (after left-multiplying by B^j and using BA = 1)
then
BA=1 => (AB-1) A^n = 0 for n>0

thus 0 = (AB-1) P(A) = (AB-1) P(0) => AB=1

Such a proof holds more generally in any ring where A satisfies
a polynomial equation P(A) = 0 such that the coefs of P commute
with both A and B, and the lowest degree coef of P is cancelable.

In essence one is employing the Euclidean algorithm to invert
A modulo P(A); this inverse is a poly in A so commutes with A
(same for any rational function R(A)/Q(A) with Q coprime to P).

Notice the natural path of abstraction from a matrix to a linear
operator to its associated (polynomial) operator algebra - where
we recognize the applicability of the Euclidean algorithm. Many
results in linear algebra generalize beautifully in this manner
and are unified in the study of modules over a PID (e.g. see
Jacobson's Basic Algebra I where he develops in such a manner
the structure theory of finitely generated abelian groups and
canonical forms for linear transformations). For a much more
comprehensive approach see Fuhrmann's textbook [1] which, as
the reviewer concludes, is unique, of unconventional beauty,
and one of the best advanced texts on linear algebra on today's
merciless market (esp. see the final 2 paragraphs of the review).

An even deeper application of the operator theoretic approach is
found in Rota's Umbral Calculus, e.g. see Steven Roman's book.

Yet again, the lowly polynomial rises? to the occasion (see
the URL below); I question "rises" because here one employs
*non-linear* (polynomial) algebra to conquer *linear* algebra!
http://www.dejanews.com/dnquery.xp?QRY=dubuque%20lowly&groups=sci.math&ST=PS

-Bill Dubuque

[1] Fuhrmann, Paul A. (IL-BGUN)
A polynomial approach to linear algebra. Universitext.
Springer-Verlag, New York, 1996. xiv+360 pp. ISBN 0-387-94643-8
MR 98a:15001 15-01 (12D10 13B25 26C10 30E10 47-01 93-01)
http://www.ams.org/mathscinet-getitem?mr=98a:15001

The book differs from other texts on linear algebra in the choice of the
material and in the approach to the classical core of linear algebra.
The hero of the book is the shift operator, which indefatigably does wonder
after wonder. After the shift operator's grand entrance in the derivation
of the Jordan form, it reappears in connection with Hankel and Bezout
matrices, it is continuously encountered in the chapter on root location,
and finally, after its metamorphosis through companion matrices into
realizations of rational functions, it becomes the main actor behind the
scenes of linear system theory.

Let F_n[z] stand for the polynomials of degree at most n-1 with
coefficients in F . Given a polynomial q in F_{n+1}[z] , the shift
operator S_q is the operator on F_n[z] which sends a polynomial
f(z) to the remainder of zf(z) modulo q(z). The structure of S_q
is determined by the prime factorization of the polynomial q(z).

Thus, to understand S_q we first have to know something about polynomials.
This reveals one of the methodological difficulties of the author's
approach: algebra must precede linear algebra. Accordingly, the first 30
pages of the book include in condensed form the definitions of and results
on the following concepts: group, normal subgroup, ring, ideal, principal
ideal, coprimeness, Euclidean ring, irreducible polynomial, integral domain,
field, quotient ring and field, stable rational function, formal power
series, module. Only after all that, which is actually the material of a
one-term course, the reader is acquainted with the notion of linear space!

Chapters 2 to 4 deal with linear spaces, determinants, and linear
transformations (= operators). The presentation is original, we encounter
polynomials constantly, and Lagrange interpolation, Taylor expansion, or
Sylvester's resultant are nice gems of these chapters. On the other hand, a
certain hurry to come to his beloved shift operator makes the author forget
to define such fundamental things as multiplication of a column by a matrix
or multiplication of two matrices (although these operations are used
throughout).

Chapters 5 and 6 are devoted to the structure theory of (linear and finite-
dimensional) operators. After introducing the shift operator S_q and
exhibiting its basic properties (with attractive interludes about circulant
matrices, Hermite interpolation, the Chinese remainder theorem, and
reproducing kernels), it is shown that every cyclic operator is similar to
S_q for some (unique) q and that an arbitrary operator decomposes into
a direct sum of cyclic operators. This gives us the Jordan form. These two
chapters are a delight for insiders, but they contain a few hurdles for
beginners [...].

After a chapter on inner product spaces, we enter the grandiose second half
of the book. Chapter 8 contains much more than its mousy title "Quadratic
forms" promises. The author treats the standard topics, such as Sylvester's
law of inertia, and then gives a really brilliant introduction to Hankel
and Bezout forms and matrices. Bezout matrices arise from representing a
rational function g in the form g = p/q with polynomials p,q, while Hankel
matrices emerge from the expansion g(z) = sum_j g_j z^-j at infinity. Both
types of matrices can be expressed in terms of shift operators, which
yields plenty of beautiful connections between them. The themes considered
in Chapter 8 include: Kronecker's theorem, Barnett factorization, Gohberg-
Semencul-Trench formulas, representation of resultants via Bezoutians,
(fast) inversion of Hankel matrices, continued fractions and the Frobenius
theorem on the signature of Hankel matrices, the Cauchy index and the
Hermite-Hurwitz formula for the signature of Hankel and Bezout matrices.

The subject of Chapter 9 is the location of the roots of algebraic
equations. It is based on ideas by M. Krein and Naimark and thus relates
problems about root location (in particular, stability, i.e. location
of the roots in some half-plane) to properties of Hankel and Bezout
matrices. Of course, the Hurwitz stability criterion is also proved.

Chapter 10 is a very readable introduction to linear system theory. The
reviewer agrees with the author in feeling that this field must be considered
as an essential part of linear algebra. Fuhrmann hits the point with writing
that "In fact, the notions of reachability and observability, introduced by
Kalman, fill a gap that the notion of cyclicity leaves open". The role played
by polynomials and the shift operator (in its disguise as a companion matrix)
in the structure theory of linear operators is performed by rational functions
and their realizations in system theory. The chapter embarks in detail on the
realization of rational functions, deals with internal (!) stabilization of
SISO systems, and also contains the Youla-Kucera parametrization. The reviewer
had to teach linear system theory seven years ago and regrets that such an
introduction to the topic as the one by Fuhrmann was not available at that
time. If the reviewer had to teach linear system theory today, he would
unconditionally base the course on Fuhrmann's exposition.

The final chapter, Chapter 11, is on Hankel norm approximation (= AAK
theory). In a sense, it deals with the approximation of rational functions
by functions with a prescribed number of poles in a half-plane. Hankel
operators again play a crucial role. The reader is acquainted with (the
rational versions of) the Beurling, the Adamyan-Arov-Krein, and the Nehari
theorems as well as with Nevanlinna-Pick interpolation.

As for the "polynomial approach", Fuhrmann writes: "The study of a linear
transformation in a vector space via the study of the polynomial module
structure induced by it on that space already appears in B. L. van der
Waerden's Modern algebra [Bd. I, Springer, Berlin, 1930; JFM 56.0138.01;
English translation, Ungar, New York, 1949; MR 10, 587b; Bd. II, Springer,
Berlin, 1931; JFM 57.0153.03]. Although it is very natural, it did not become
the standard approach in the literature, most notably in books aimed at a
broader mathematical audience, probably due to the perception that the study
of modules is too abstract." Clearly, we all feel that there does not exist
"the standard approach" to linear algebra. This book is an attempt to make
polynomial models "a standard approach" to the piece of linear algebra
creeping around the Jordan form, but the reviewer is nevertheless pessimistic
whether such methods will ever conquer the first-year courses.

In summary, the approach pursued by the author is of unconventional beauty
and the material covered by the book is unique. The book is certainly not
an ideal text for rank beginners. However, persistent students with some
previous training in linear algebra and unprejudiced professionals with
some inclination to operator theory will appreciate this book as one of
the best advanced texts on linear algebra on today's merciless market.

Reviewed by A. Bottcher