Domino effect

 

Do I agree with the above? Mostly - yes. But not always. For instance at the end of the last post I told my Reader what I intend to do in the next post. But I did not show it FIRST. I am showing it only now.

But in general it seems to be true that we often tend to disperse our energy on telling the world what we intend to do: 

"World, you will see how great things I will do! I will do THIS and THAT. The day will come when you see and realize how great I am!"  

And then we have lie to ourselves when we realize that, in fact, our true destiny is in doing something else. 

Anyway yesterday became today and today I will show how to get nice expressions for C and D of the previous post. This is its continuation.

Let Z be mxn  (m,n >0) complex matrix. So Z is, in general, rectangular, not necessarily square, matrix. Let Z* be its hermitian conjugate (complex conjugate transpose). Then Z* is nxm. Moreover ZZ* is mxm, while Z*Z is nxn. We will denote by I the unit matrix, whether it is mxm or nxn,  will depend on the context.

We first notice that 

ZI = IZ

Now we will use the associativity of matrix multiplication:

Z(Z*Z) = (ZZ*)Z

Z (Z*Z)(Z*Z)=ZZ*ZZ*Z=(ZZ*)(ZZ*)Z

and in general, for any n >= 0

Z(Z*Z)ⁿ =(ZZ*)ⁿZ

This is our "domino effect" from the title of this post: we push from the left with Z and Z falls down on the right. In between powers of Z*Z get replaced by the same powers of ZZ*.

Thus for any analytic (representable as a convergent power series) function f of a complex variable we have

Z f(Z*Z) = f(ZZ*) Z 

Similarly

Z* f(ZZ*) = f(Z*Z) Z*= 

Nice? Nice!

Notice that we are taking powers of square matrices. Taking powers of a non-square matrix would not make sense!

In the previous post we have obtained the following formula for B:

B = 2(I-ZZ*)^-1 Z

We also know that C=-B*. We can now use the domino effect formula to obtain

C = -(2(I-ZZ*)^-1 Z)*=-2Z*(I-ZZ*)^-1 =-2(I-Z*Z)^-1 Z*

So

C = -2(I-Z*Z)^-1 Z*

It remains to calculate D. In the previous post we have obtained:

(5) D = (I_n+B*B)^1/2

and 

B = 2(I-ZZ*)^-1 Z

Thus 

B*B = 4Z*(I-ZZ*)^-2 Z

Using the domino effect formula we can rewrite it as

B*B = 4(I-Z*Z)^-2 Z*Z

Now I+B*B can be easily calculated to give

I+B*B=(I-ZZ*)^-2 (I-ZZ*)²  + 4(I-Z*Z)^-2 Z*Z = (I-ZZ*)^-2 (I+ZZ*)² 

and therefore 

D = (I-ZZ*) (I+ZZ*) 

 P.S.1. Here is a continuation of P.S.1 from the post Eine Klein Al Gebra . My comments on the book "Mistakes we made: But not by me" by Carol Tavris and Elliott Aronson. In a chapter "Cognitive Dissonance: The Engine of Self-Justification" the authors give us a serious warning: how easily we fall into self-justification! For instance a smoker will try to find all possible (often irrational, neglecting completely rational ones) arguments to convince themselves and others that smoking is good for you. Or take this argument from the book:

>Some scientific evidence for the power of irrevocability comes from a clever study of the mental maneuverings of gamblers at a racetrack. The racetrack is an ideal place to study irrevocability because once you’ve placed your bet, you can’t go back and tell the nice man behind the window you’ve changed your mind. In this study, the researchers simply intercepted people who were standing in line to place two-dollar bets and other people who had just left the window. The investigators asked them how certain they were that their horses would win. The bettors who had placed their bets were far more certain about their choice than the folks waiting in line. Yet nothing had changed except the finality of placing the bet. People become more certain they are right about something they just did if they can’t undo it.

While the above is certainly true enough, it is dangerously only partially true. Here is why: a person that decides to quit smoking must often show a very strong will power to convince himself constantly that the decision was right. Writing down arguments supporting this decision and rereading them again and again may be of help.

>Another example from ""Narcissistic Personality Disorder How to Spot the Subtle Signs of a Narcissist and Continue to Thrive After an Encounter" by Tony Sayers:

George had just about had enough of his father’s behavior. He didn’t like being belittled, torn down, and compared to his dad, and he hated having his hard work underappreciated simply because he was ‘just his father’s son.’ So, after their last heated argument, he decided to just walk away and leave their relationship at that. Whether they’d ever be on good terms again, he was uncertain. But he was happy to finally be free from his dad’s abuse.

In the weeks following his falling out with his dad, George started to feel a strong sense of isolation and guilt. He felt as though he had wronged his father, and struggled to resist the urge to reconcile, knowing full well that it would only give his dad the fuel he needed to make George feel bad about protecting and defending himself.

Here we have again: a difficult but necessary decision. Also in such cases self-justification is not a bad thing. It is a ncessity. Similar examples can be found, for instance in "Energy Vampires. How to protect yourself from toxic people with narcissistic tendencies" by the same author and in "The Borderline Personality Disorder. Survival Guide" by Alexander Chapman and Kim L. Gratz. In similar cases taking a difficult decision and keeping to it is the only solution to otherwise never ending problems with constant ups and downs.

P.S.2. As I am recently (encouraged by Laura)  studying Christopher Langan's  "philosophy", here is something related: 

The Results & Features of a Person with a High IQ | Jordan Peterson