Eine Kleine Al Gebra

 From Wikipedia:

Algebra (from Arabic ‏الجبر‎ (al-jabr) 'reunion of broken parts

And it is time for starting such a reunion. 

It is said:

Matthew 9:17 King James Bible. "Neither do men put new wine into old bottles; else the bottles break, and the wine runeth out, and the bottles perish; but they put new wine into new bottles, and both are preserved."

I will be putting old wine into old bottles Algebra is old and conformal group that includes similarity transformations (as above so below) is probably even older.

Quod est superius est sicut quod inferius, et quod inferius est sicut quod est superius.

That which is above is like to that which is below, and that which is below is like to that which is above.

So here it comes: Eine Kleine Al Gebra. But first the old music-wine in old bottles

Allegro

Let m,n ≧ 1 be two integers. Let X be a complex vector space of complex dimension m+n. 

Let X be equipped with a fixed sesquilinear form (z,z'), antilinear in the first argument, linear in the second one.

We assume that X admits a basis e₁,...,e_m,e_m+1,...,e_{m+n  such that, with respect to this basis, the scalar product (z,z') takes the form}

(z,z') = -z₁* z'₁ - ... -z_m* z'_m + z_m+1* z'_m+1 + ... + z_n* z'_n

where the star * stands for the complex conjugation. Such a basis will be called orthonormal. We say that (z,z') is a hermitian scalar product of signature (n,m). Notice the order: (n,m). n plus signs, m minus signs.

Any two orthonormal bases are related by a transformation from the group U(n,m). We will write matrices of U(m,n) in a block form

U =

A  B

C D

where the matrices A,B,C,D are repectively mxm,mxn,nxm,nxn.

Given an orthonormal basis X can be identified with C^m+n.

Let † denote the standard hermitian conjugate (conjugate transpose). Then, (z,z') can be written as

(z,z') = z^†J₀z'

where J₀ is the diagonal (m+n)x(m+n) matrix J₀= diag(-I_m, I_n) and I_m, I_n are the mxm and nxn unit matrices respectively. The group U(n,m) is the set of all matrices U satisfying

U^†J₀ U = J₀

Writing U in a in block matrix form as above, the condition for U to be in U(n,m) translates into:

A*A - C*C = I_m, D*D - B*B = I_n

A*B - C*D = 0, B*A - D*C = 0

In the following, for the ease of notation,  we will use * to denote the ordinary hermitian conjugate for mxm,mxn,nxm, and nxn matrices.

We notice that A*A = I_m + C*C and C*C is non-negative. Therefore A*A ≧ I_m, and, in particular, A is invertible. D is invertible for a similar reason.

From Wikipedia:

In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V

We will be interested in in a subset of Gr(n,X), Namely we will be interested in the set J of all  n-dimensional subspaces of X on which the scalar product (z,z') is positive definite. Choosing an orthonormal bases one such subspace is evident: it consist of column vectors u,v, u from C^m. v from Cⁿ. for which u=0.

It is more or less evident that U(n,m) transforms J into itself and that the action of U(n,m) on J is transitive. The stability subgroup is U(m)xU(n), therefore J can be identified with the manifold of equivalence classes (cosets) U(n,m)/(U(m)xU(n)).

We will need an explicit parametrization of J.

Let us do some Christmas Eve talk. Say V is an n-dimensional subspace of X on which our scalar product is positive definite. It is then necessarily a maximal subspace with this property, and that because our scalar product is of signature (n,m). 

Let W be the orthogonal complement of V. Then on W the scalar product is necessarily negative definite. Let E be the orthogonal projection on V. Then E*=E and EE=E. Here, and in the following, * applied to linear operators acting on X will denote the hermitian conjugate with respect to the scalar product (z,z') on X. The eigenvalues of E are 1 and 0. 1 on V and 0 on W.

Then F=I-E is the orthogonal projection on W - the orthogonal complement of V. We also have F*=F and FF=F. The eigenvalues of F are 0 and 1. 0 on V and 1 on W. Subspaces V and W are mutually orthogonal.

Let us define 

J=E-F.

Then J=J* and JJ=I. The eigenvalues of J are -1 and 1. 1 on V and -1 onW.

Let us define a sesquilinear form (z,z')_J = (z,Jz'). We claim that (z,z')_J is positive definite. Indeed, let z be any nonzero vector in X. Then z decomposes into z=v+w, where v is in V and w is in W. Thus Jv=v, Jw=-w. But then (z,z)_J=(z,Jz)=(v+w,J(v+w)=(v+w,v-w)=(v,v)-(w,w). But if z is not zero, then v or w (or both) must be non-zero. For a non-zero v, (v,v) is positive and for a non-zero w, -(w,w) is positive. Therefore (z,z)_J is positive. 

This way with each n-dimensional subspace V of X on which the scalar product is positive definite we have associated a linear operator J on X such that 

1) J=J*, 

2) JJ=I, 

3) and (z,Jz') is positive definite. 

Let us see that the converse is also true. Namely, Let J has the above properties. Define E=(I+J)/2, F=(I-J)/2. Then, because of 1) and 2), we have that E*=E, EE=E, F*=F, FF=F, EF=FE=0, E+F=I. E and F are two complementary orthogonal projection operators. Let V be the subspace belonging to eigenvalue 1 of J, W be the subspace belonging to the eigenvalue -1 of J. E projects on V, F projects on W. Moreover, if a non-zero v is in V, then Jv=v and, because of 3),  (v,v)=(v,Jv)>0. Similarly, if a nonzero w is in W then Jw=-w and, because of 3),  -(w,w)=(w,-w)=(w,Jw)>0. Thus V is maximal positive and W is maximal negative subspace of X. QED.

Our next step is parametrization of such operators J. We will do that in the next post. Here we just note that these J are very special elemenets of the group U(n,m) (Why so?).  We will cal them "symmetries".

Merry Christmas!

P.S.1 I am slowly (mainly during breakfasts) reading the book "Mistakes we made: But not by me" by Carol Tavris and Elliott Aronson. 

As I have promised, here are my first initial comments on the book "Mistakes we made: But not by me" by Carol Tavris and Elliott Aronson. Today I will deal only with the Itroduction and with quite general comments.

It starts like that:

INTRODUCTION

Knaves, Fools, Villains, and Hypocrites:

How Do They Live with Themselves?

Mistakes were quite possibly made by the administrations in

which I served.

—Henry Kissinger, responding to charges that he

committed war crimes in his role in the United

States’ actions in Vietnam, Cambodia, and South

America in the 1970s

And a liitle further on we find the following biting remark:

When Henry Kissinger said that the administration in which he’d served may have made mistakes, he was sidestepping the fact that as national security adviser and secretary of state (simultaneously), he essentially was the administration. This self-justification allowed him to accept the Nobel Peace Prize with a straight face and a clear conscience.

So far so good. There are many more illustrative examples of self-justification in the introduction. But then there is a pragraph that have rised a red frag in my mind. Here it is:

By understanding the inner workings of self-justification, we can answer these questions and make sense of dozens of other things people do that otherwise seem unfathomable or crazy. We can answer the question so many people ask when they look at ruthless dictators, greedy corporate CEOs, religious zealots who murder in the name of God, priests who molest children, or family members who cheat their relatives out of inheritances: How in the world can they live with themselves? The answer is: exactly the way the rest of

us do.

It is evident to me that the authors neglect here important discoveries about workings of human minds: not all brains are wired the same way. If we read carefully „Inside the Criminal Mind” by Stanton E. Samenow, we learn that

„There are people who would be “criminals” no matter where they live. These are individuals for whom to be someone in life is to do the forbidden, whatever the forbidden might be.

…

I submit that there is a major difference between the person who tells an occasional lie and an individual who lies as a way of life. The criminal lies to cover his tracks (he has a lot to conceal) and to get out of a jam that he has created for himself.

However, he also lies about the most minuscule matters even when there is no ostensible reason. He’ll say that he went to one store when he really went to another. Some people in the mental health field will conclude that this is pathological or compulsive. This is not the case. The lie that makes no sense does make sense when you understand the mentality of the liar.”

Not all men are wired the same way. Some are wired differently. And not all mistakes are equal. Results do count. A politician has more responsibilities than a receptionist in an inn. There is a difference between a mistake that costs human lifes and a mistake that has no serious  consequencesat all. It is therefore important to know ourslves, self-observe, consciously compensate for all defects in our wiring. And take responsibility for all results of our actions, and for our negligance of actions, when such are needed.