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Mar 16, 1999, 3:00:00 AM3/16/99

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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

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.

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

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

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

: 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>

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.

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?

>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.

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

>>

>>: 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>>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.

>>

>> 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.

>

>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.

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).)

>

> 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".

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.

>

[...]

>

> 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 = != == (!)

-----------------------------------------------------

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".

|>

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

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

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

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