Thursday, April 27, 2017

Math Revisited: Linear Algebra (Dimension)

The concept of 'dimension' in linear algebra basically has direct applicability for all finite dimensional vector spaces, up to some limit, n. There also occur infinite dimensional vector spaces for which it's possible to provide an infinite basis, but these are more peculiar to specialized applications as in advanced quantum theory (e.g. Hilbert spaces as infinite dimensional vector spaces). We will eschew this special version until a much later time, next year, when we survey some topics in advanced quantum mechanics.

Meanwhile, if V is some vector space having a basis consisting of 'n' elements, we say that n is the dimension of V.

If V consists of 0 alone, then V does not have a basis, and one says V has dimension 0.

As another generic example, let the vector space R    have dimension n over R, and the vector space C   have dimension n over C. More generally, for any field F, the vector space F  has dimension n over F. One can therefore say that the n vectors:

(1, 0.........), (0,1...........0), (0, 0, 1..........0),.........(0, ............0, 1)

form a basis of  Fover F.

Then the dimension of the vector space V over F is denoted by: dim_F V or more simply, dim V.

Associated with these considerations is the concept of a maximal set of linearly independent elements of a vector space. Then let: v1, v2, be linearly independent elements of a vector space V, then the elements w, v1, v2, are linearly independent and v1, v2, for a set of maximally independent elements.

Ancillary or Auxiliary Theorems:

1) Given V is a vector space and one basis has m elements and another basis has n, then m = n.

2) Let V be a vector space and {v1, v2, } be a maximal set of linearly independent elements of V, then {v1, v2, } is a basis of V.

3) Let V be a vector space consisting of n elements. Let W be a subspace which does not consist of zero alone. Then W has a basis and the dimension of W is less than or equal to n.

4) Let V be a vector space over the field F and let U,W be subspaces. If: U + W = V and if U/\W = {0} then V is the direct sum of U and W. (Note: /\ denotes intersection)

5) If V is a finite dimensional vector space over F, and is the direct sum of subspaces U, W then:

dim V = dim U + dim W


1) Let V = R2 and let W be the subspace (2,1). Let U be the subspace generated by (0, 1). Show that V is the direct sum of W and U.

2) Prove theorems (1) - (4)

3) what is the dimension of the space of 2 x 2 matrices? Give a basis for this space.

Parsing The "High Conflict Personality" - Does Trump Qualify?

Evidently, there's a new subclass of psycho that has been flying under the radar and which needs to be integrated into the maelstrom of mentally unstable misfits, including malignant narcissists and authoritarians. That is, the "high conflict personality".   According to Bill Eddy, one of the 'high priests' of this specialty area:

"Narcissistic high conflict people like being unpredictable and beholden to no one. They hate rules and feel superior to them. So being consistent feels confining and they generally like to keep everyone else guessing. It’s rule by personality, not by logic or law. Sociopathic high conflict people will say (lie) and do whatever is convenient or appealing at the time, without conscience or consistency. They like instability because it keeps everyone else off-balance. This especially can be demonstrated with a lack of loyalty to his associates, discarding them at will and turning his attention to new shiny objects and people (wives, for example)."

This brand of looneytune sounds like Donald Trump and with good reason: it basically nails his unstable, predatory shtick to a tee. The question is: Can such an unbalanced person govern with any degree of efficacy - and the answer nearly 100 days into this misbegotten presidency appears to be a resounding 'no'.  Nothing Trump has done thus far, from his half-assed, juvenile "executive orders", to his impetuous cruise missile strike on Syria, to his bellicose rhetoric in confronting Kim Jong Un over the North Korea issue, shows he possesses the leadership mettle to inspire any confidence. He is more a buffoon, and totally ignorant at that. Hence, his only ploy or play is to try to keep others off balance. Or to use the phrase of one HuffPost contributor: "A mad monkey with a gun loose on the street".

He does this by constant appeal to conflict.  Is this a dangerous and demented shtick? Yes it is. According to Bill Eddy who wrote the definitive book on high conflict personalities,  Trump is  “…potentially the most dangerous person in politics since Adolf  Hitler.”  Got that? The most dangerous politico since Adolf Hitler.

 In an interview with a San Diego journalist,  referencing the quote, Eddy was asked: "Clearly a terrifying opinion — yet we do remember the ardent hordes of Hitler’s followers. Can you elaborate?"

Eddy replied:

"There are similarities regarding how he connects with his followers, so that they become more passionately connected when he and they are criticized. It strengthens their bond. It’s a major part of my book. In a nutshell, it’s because he uses emotions instead of logic, and uses repetition on a greater scale than most politicians since Hitler. He has conditioned his followers to his simple arguments from day one. Hitler was known for spewing ten times as many words as any other politician, and he used modern media for voice (daily speeches  on the radio) and face (movies of him speaking at his huge rallies). These emotional media projecting face and voice are much more powerful on our brains than printed words."

In other words, Trump has gone for the "lizard brains" of his doting followers and enslaved them wholesale. This is something that neuroscientist Robert Ornstein once raised, in his book, 'The Evolution of Consciousness' and later works, on how easily the limbic system is activated and overtakes reason. In fact, we are talking about an entire brain complex, referred to as the "R-complex" - for example by Carl Sagan in his book, 'The Dragons of Eden'.

The main aspect to note concerning the R-complex and its dynamics is that primitive emotion rules, and logic is far removed from any integration. It's as if reason doesn't exist. This is also why it's impossible to get through rationally to the Trump groupies and voters. Their brains are pitched to the emotional centers of the R-complex not to the neocortex higher thought centers. You could as well be
arguing with a Komodo Dragon.

It's unsettling enough to find these properties in ordinary citizens, but  absolutely terrifying for them  to be associated with leaders whose fingers are on the nuclear football.  This is because the lizard brain, i.e. R-complex, doesn't weigh responses in proportion to stimuli. If buzzed by a fly it will reach for a sledge hammer, and if a hornet's nest has been built on a deck it will take out the flame thrower. If  the house burns down "them's the breaks".  This is why having an unhinged "high conflict" nut like Trump at the helm for this Korean crisis ought to keep everyone up at night - even the fool who scribbled a recent NY Times piece to the effect there's nothing to worry about.  "Despite the showdown there won't be a war."  Will you place a bet on that, fool?

When the same SD journalist asked Eddy:  "Do you really believe that, per his leadership, 'sooner or later we will have a war on our hands?'”

Eddy replied by quoting a statement from his book:

“He tends to inspire violence and lack of restraint — which leads to lack of physical restraint, which leads to organized aggressive behavior — which leads inevitably to war. He will “split” the world into allies and enemies. We will have more chaos than we do now in the Middle East — and on American soil. Friends and family members will start hating each other, and school children will become disrespectful and violent towards people who look different from them.”

When  Eddy was asked if we ought to scared, that is those of us with functional higher brain centers, he replied:

" Yes. Moderate Republicans  may limit his damage and keep him tied up in procedural  knots but it won't be enough."

What we need is the Dems to take over the House next year and begin impeachment proceedings as soon as feasible. It is clear we can't depend on the Reepos to do their jobs. They have too much invested in this maniac to help push through their radical agenda. Stay tuned.

Wednesday, April 26, 2017

Math Revisited: Linear Algebra (3)

We now look at orthonormal bases. The term sounds esoteric but many would have encountered it before either in general physics or in advanced (AP) physics courses taken in high school. This would be in conjunction with the dot product of vectors, such as illustrated in Fig. 1.

Basically, in some Euclidean (straight line, 3D) system of coordinates, two vectors are considered orthogonal if their inner product is zero, as shown. The geometric properties here are assumed to be on the basis being orthonormal, i.e. composed of pairwise perpendicular vectors with unit length. Thus, in the figure shown, vectors A and B meet this condition (and the computation is shown for the vectors A, B) as do the vectors B and C. Then the vectors A, B, C meet the condition for an orthonormal basis. There are proofs available but those are beyond the scope of this blog.

Now, in applications of this concept, what the student is usually asked to do (say in his linear algebra course) is find an orthonormal basis for a "subspace" of R 3  (e.g. applied to a Cartesian space of three dimensions) which is generated by specified sets of vectors.


Find an orthonormal basis for the subspace of  R 3 generated by the vectors (1, 1, -1) and (1, 0 ,1).

We let A = (1, 1, -1) and B = (1, 0, 1)

The orthonormal basis for A is just:

A/ ‖A ‖ = (1, 1, -1)/ {1+ 1  + (-1) 2 } =    (1, 1, -1)/  Ö 3

The orthonormal basis for B is:

B/ ‖B ‖ = (1, 0, 1)/ {1  + 0   + (1)} =

(1, 0 , 1)/ Ö2

Of course, the beauty of linear algebra is that it can be generalized to Euclidean spaces beyond the mundane 3, hence we can look at subspaces in  R 4, generated by sets of vectors (v1, v2, v3, v4).


Find an orthonormal basis for the subspace of R 4 , generated by the vectors:

A = (1, 2, 1, 0) and B = (1, 2, 3, 1)

For A we have the orthonormal basis:

A/ ‖A ‖ = (1, 2, 1, 0)/ {1 +  2  + 1 +  0 } =

(1, 2, 1, 0)/ Ö6

For B we have:

B/ ‖A‖ = (1, 2, 3, 1)/ {1 2 + 2 2 +  3 2 + 1 2} =

(1, 2, 3, 1)/Ö 15

Practice Problems:

(1) Find the orthonormal basis for the subspaces of  R 3  generated by the vectors:

A = (2, 1, 1) and B = (1, 3, -1)

(2) Find the orthonormal basis for the subspaces of  R 4 generated by the vectors:

A = (1, 1, 0, 0)

B = (1, -1, 1, 1)

C = (-1, 0, 2, 1)

(3) Find an orthonormal basis for the subspace of the complex space  C 3 generated by the vectors:

A = (1, -1, i)


B = (i, 1, 2)

Thoughts On The Closure Of ALL EXPERTS

All Experts is the free question and answer service that for 19 years provided users with answers to their questions in a vast variety of subject areas from biology, medicine, geology, religion, politics, and psychology to astrophysics and physics.  Since late 2004 I participated in the Astronomy forum - this before there was All Experts (it was known then as 'Experts Central') then three years later joined the astrophysics expert forum, as well as the atheist -agnostic forum.

As per a notification sent out a week ago, the site shut down from April 24th. Anyone trying to aces it via Google or the url will simply be shown a terse message noting the 19-year run and that "more than two million" questions were answered.

Why the closure, we can't say. Experts were simply sent the brief notice with no further explanation. One reason is that perhaps there wasn't the money available to support the financial maintenance and even a putatively "free site" needs money to operate - if only to continue to support nearly 500 different forums, responses and further exchanges.

But let's be clear the site wasn't always smooth sailing and for experts often a source of frustration. After eight years answering astronomy questions, for example, I had to go to the option of  "no ratings allowed"  - ratings being given in the several areas of: politeness, knowledge, clarity of response, and timely response.    What constantly irritated me was how so few questioners knew how to give an appropriate rating, especially to knowledge and clarity of response.  For example, they'd deliver a '10' in knowledge e but then an '6' in clarity of response. But how would one be cognizant that the knowledge aspect was as high as 10 when they believed the clarity was only at a 6? It was preposterous.  

I tried to attach the proviso that if the answer still appeared to be abstruse, or too difficult, the questioner could reframe the question or make it more specific. (That was another gripe - often the questions were so poorly worded one couldn't make any sense of them and had to ask the person to please rewrite it, try to be more specific or to clarify a term used).   Often there was no further question, so the expert is left with just the crummy rating - and bear in mind we aren't getting paid a damned cent so the ratings comprise our only "earnings".

I retained the ratings for the astrophysics largely because I'd become a lot more rigid in what categories of question I accepted. If it seemed that bare balderdash or ignorance was the theme I delivered a polite rejection response.  Or, if there was the slightest divergence from what astrophysics included. For example, questions to do with meteorology, say the formation of "Moon dogs" or "halos"  forming around the Moon.   While perhaps making it as an astronomy question - ice crystals forming in the atmosphere - it doesn't make the cut for astrophysics. 

As I made clear in subsequent rejection letters the topics for astrophysics included: star formation and evolution, the HR diagram, stellar models, stellar spectra, star cluster dynamics, galactic cluster properties, Hubble diagram, expansion of the universe, solar flares, sunspot properties and coronal mass ejections as well as prominences, plasma waves and instabilities. 

Apart from the poorly worded or misdirected questions there were the many from crackpots. These were most often poorly or semi-educated people who were perhaps above average in intelligence, had a bit of a "brain storm" and fancied themselves a new Einstein for having found some seemingly novel relationship. One of these was a guy in his fifties or sixties, admittedly having "little education" who came up with the notion of "Earth flares", i.e. analogs to solar flares but occurring on Earth.

After several fruitless exchanges where I emphasized there was no observational basis for his entity, namely any magnetic signature for rapid release, he finally agreed to go to another expert. He just couldn't admit his own "thesis" was cockeyed and - in essence - pseudo science.

Another pair of crackpots (from the Middle East) insisted they had overturned Kepler's 2nd law of planetary motion, the "equal area" law.  One of them actually wrote:

"Kepler's area law says r*Vp=Ct.  Newton's universal attraction force says this force is radial F=Fr and a perpendicular force (Fp)  to the radial, also a side force component does not exist.  So: m*dVp/dt=Fp=0.  Then dVp/dt=0 and with integration we get Vp=Ct.     f Vp=Ct is correct, elliptical orbits theory has to be modified to a new theory, new math. And the motion equation should be r=-4*t^2+4*t*T-4*T^2/6 .This equation does not indicate an ellipse but a parabolic vortex spiral."
For his benefit I made several points:
1) He never expressed the 2nd law in proper form and indeed it makes no sense at all as portrayed.

2) He never defined 'C', and whether it is the same as h .  He doesn't say so one can't just assume it.

3) His equation m*dVp = Fp = 0 is incorrect. It should read:
(m1) d2   r1/ dt 2 =  G m1m2 (r^)/ r 3   = (m2) d2   r2/ dt
Hence he failed to distinguish between a real force and a non-existent force, i.e. perpendicular to the orbit (since the gravitational force of interaction acts through the mass centers) so mg = 0 i.e. weightlessness applies.
In effect, his integration result, e.g. Vp=Ct  is spurious .
After much back and forth in the rejection regime he rendered a final assertion he believed would force me to go to his side, writing:
"Anyhow,  sending celestial probes to the Moon,or Mars or to any body will still be successful even if the orbits are triangular. It does not depend on  the form of the planet's trajectory . It is controlled from the earth."

In other words, he's saying you just need to fire the rocket into space, and the human controllers on Earth (like at Johnson Space Center) will do the rest,  to "steer" it to the destination.    I had to summon all my will power to avoid calling him a total nutcase, but again we must allow common courtesy even for kooks.
Then there was the Canadian "Bree"  (and several others with different identities or names) who sought to browbeat me into accepting a novel presentation of units which "proved" one or more of her crackpot ideas. She wrote, as "mathematical proof":

1)     Speed of Light  =  orbit Velocity^2  /  Pi       (Enormously Important Equation)

2)     Speed of Light  =  Acceleration of Gravity  x  orbit Time

3)     Speed of Light  =  Acceleration of Gravity  x  (earth Volume  /  orbit Circumference x Tau2)

4)     Speed of Light  =  Acceleration of planet earth in orbit  x  Diameter of orbit

5)     Speed of Light  =  Density of earth  x  orbit Velocity x 21  
None of which made any sense in terms of actual physical units and dimensions as I tried to show her, to no avail.  Her baloney was a perfect example of similar piffle cited by Charles Seife in his superb book:  'Proofiness: How You're Being Fooled by the Numbers.'   Seife decried the tactic of using numbers not just to lie but to baffle the susceptible and gullible with bullshit.  He referred to a common failing of most people unversed in math to be hoodwinked merely because some form of math or numbers are interjected into arguments.  Not just using numbers to bolster one's argument. In his words, to use fake numbers to prove falsehoods and to seek to prove something is true - even when it's not- is one of the most egregious forms of  intellectual  fraud. But "Bree" believed herself to be an incomparable genius of the first order.

The first thing any first year Physics student learns - or should - is to check a formula or claimed relationship for consistency of the physical units used. If their combination doesn't yield any recognized physical quantity - defined as part of the standards and constants published yearly by the American Institute of Physics - then the results are balderdash, pure and simple. Let's take her example (4) where:

Speed of Light  =  Acceleration of planet earth in orbit  x  Diameter of orbit

What are the units here? The speed of light c is in meters per second, or m/s. The acceleration a is given in terms of  m/ s 2  and the diameter of any orbit (any orbit) is a length -  so given in meters (m). Then multiplying them together one gets:

(m/s)  =  ?  (m/ s 2) (m)  =  m2 / s 2

Which is emphatically not the same as the units for velocity of light! Let's also check her "enormously important equation" (1): Speed of Light  =  orbit Velocity^2  /  Pi      

Again, we examine the product of the units to see if there's a physically meaningful quantity that results or if this girl is just seeing things, or seeing what she wants to see.  We have the speed of light on one side (units m/s ) and on the other the orbit velocity squared:

(m/s)2    =  m2 / s 2

Note that the presence of pi is neither here nor there as it's dimensionless.       "Bree" leaves herself open on so many fronts it's essentially laughable, and in ordinary discourse she'd be put down as a dummy or dupe, but as experts  we had to strive for politeness. This irrespective of what mutation of knowledge we were confronted with. (Including one Saudi character nicknamed "Amtry" who tried to argue atheists were less than moral because they objected to have prostitutes flogged, or those women seeking abortions. 

The thing that really bothered me in all this kerfuffle with the Canuck lass(-es) was, not so much the snarkiness and bogus certitude of these young women, but their failure to recognize that the combination of units yields zilch. A squared velocity is simply NOT the same as a velocity. Both girls - if indeed there are two - would merit an F in any physics class of mine.  Worse, if I could I'd deliver an F-minus because of the lack of critical thinking.

Sadly, if this girl or girls are infected by this stupidity many others might be too. Who knows how many physics students have gone on to the website referenced and then belabored their profs with this bullshit, wasting all kinds of time?  But one thing I've learned is you can't argue with ignorant students or ignorant people who are already committed to believe what they want. In this case, because the "maths' seems to work out (at least in their heads) they believe it like a sacred book.

One would have thought they'd back off but that's not the nature of crackpots.  This was even after I had written in one rejection:.  Btw, anyone can concoct and hurl together dissimilar units to make them appear valid, but garbage in = garbage out. The sure sign of pseudo-scientific claptrap and bunkum is when one reads:

"The earth in orbit is electrical, the Tesla is present due to the earth moving through the aether.  When you multiply earth Acceleration by the Diameter as well as the hidden Tesla you get the Velocity of light and its CORRECT UNITS."

Any physics student of mine that would scrawl such drivel would get an F - for the semester! An orbit being electrical? Moving through the "ether"? The ether doesn't even exist - the Michelon -Morley experiment showed the concept of ether is redundant.  The rest is pure poppycock. The waste of a mind is a terrible ting and this nonsense is an enormous waste of a mind - as well as the minds led to buy into it

 At least 5 months passed before the following bit of bafflegab gibberish (under the header 'Sound math')  arrived in my inbox, doubtless compliments of the selfsame Canadian twit, now using the name "Brianna":

Our SOLARMath equations work with "Kilograms" ( )

But because Avogadro's number is atoms per mole which leads to atoms per "gram", prior to commencing the equation below, we need to convert the initial Mass of the earth to grams.

The Equation is;

The Mass of the earth divided by Gravity^4  x  Acceleration of earth in orbit  =  Avogadro's number.

5537831004648121688015772977.2114697 grams  /  (9.8)^4  =  600392689687827221617859.5082045622325038

600392689687827221617859.5082045622325038  x  .001003207246557  =  602318297074676474879.204

I replied with a customized rejection and was as polite as possible, emphasizing that the astrophysics forum (at least for my domain of questions) had to meet at least one of a set  of content criteria for consideration as a legit astrophysics query. She then fired back "Are you drunk? Of course it's not astrophysics, it's math!"    But any serious inquirer in astrophysics - or math- would know the multi-digit explication as shown would instantly identify  a hardcore crank -  not to mention the units being all wrong.

Then there was Steve Torchinsky  (an "engineer turned radio astronomer") who "torched" me  ("so-called astronomy expert") for advising an Indian engineering student  who wanted to transfer into astrophysics that it might be "too hard" a path to successfully negotiate. Especially going from engineering to highly abstract (relatively!) astrophysics courses like stellar modeling, evolution and plasma theory.  Torchinsky blabbered on about the hundreds who "made it" (say in radio astronomy) but never referenced my presumption that one had to get through a very difficult undergrad and graduate series of courses first. A battery that many engineers - including my mechanical engineering brother in law- have said would have been their undoing. Could he have made it? Maybe? Would he have? I don't know and neither does Torchinsky.

Apart from these issues there've always been the niggling critiques (in ratings sites)  of All Experts, i.e. they "don't know enough about my specific problem" or "they take too much time off etc. WTF do you expect for a FREE service? All of which shows me that the widespread trope that "people devalue and dismiss anything given for free" is probably true. So, as they say - let the naysayers and complainers carry  on now that All Experts is gone. I am sure Google will be able to answer all their specific questions - if they just take enough time and get through the first gazillion irrelevant pages!

Tuesday, April 25, 2017

Math Revisited: Linear Algebra (2)

We saw how one can obtain the characteristic polynomial from a matrix as well as the eigenvalues. We now want to use similar methods to show how the eigenvectors can be obtained.   To get the eigenvectors is just straightforward and merely requires obtaining simultaneous equations in x, y for example - based on using the rows in the matrix and applying each of the eigenvalues to them.

Consider for example:

A =


We have P_A(t) =

(t -1.........i)

So: P_A(t) = (t - 1) 2  - (-i)(i) = t 2 -2t +1 -1 = 0

Then: P_A(t)   = t 2  - 2t = t(t - 2)

The eigenvalues E1,2 are:

E1 = 0, E2 = 2

To get the eigenvectors is just straightforward and merely requires obtaining simultaneous equations in x, y for example - based on using the rows in the matrix and applying each of the eigenvalues to them.

For example, take E1 = 0, then the resulting equations are:

x - iy = 0

ix + y = 0

or x = -iy and y = ix

The eigenvector is easily solved for and is:

v1 =

Next, take the eigenvalue E2 = 2, then the simultaneous equations from the matrix A are:

x + iy = 2x

-ix + y = 2y

which yields: x = iy and y = -ix

Or, an eigenvector of: v2 =


Consider now:

A =

(1.. …. .2)

Then:  P_A(t) =

(t - 1......2)
2........t +2)

P_A(t) = (t - 1)((t + 2) - 4

P_A(t) = t  2 + t - 2  -4 =  t 2 + t - 6

But: t 2 + t - 6 = (t + 3) (t - 2)

So: E1 = -3, and E2 = 2

To get the eigenvector associated with the eigenvalue E1 = -3, we form the left side of the algebraic equations in x, and y using A such that:

x + 2 y = -3x

2x - 2y = -3y


4x + 2y = 0

2x + y = 0


4v1 + 2v2 = 0
2v1 + v2 = 0

and solving the simultaneous eqn. yields: v2 = - 2v1, or

v =


For E2 = 2, we may write:

x + 2y = 2x

2x -2y = 2y

Again, the eigenvalue (E2) is always multiplied by x and then y to give the column matrix
comprising the right side, with x-value on top, and y-value on the bottom. Simplifying the preceding equations:

-x + 2y = 0

2x - 4y = 0


-v1 + 2v2 = 0

2v1 - 4v2 = 0

which yields: v1 = 2v2, so the eigenvector in this case is:

v =


Working these linear algebra problems is fairly straightforward once one follows the steps such as I've outlined above.

Does Trump Have The Brains To Avoid A Nuclear Confrontation? NO!

Related imageNo automatic alt text available.
"They gotta be taught a BIGLY lesson! That punk Kim Jong Un can't get away with threatening me! I'm gonna bomb 'em back to the stone age!'

The question for the CNN panel yesterday evening was how best to respond to the continuing North Korean threats, including taking out the Carl Vinson and its strike group, while also holding an American hostage. Former New Mexico Governor Bill Richardson offered the most serious warning: "We can't treat  Kim like a normal guy. It would be a huge mistake".

These echoed the words of Denver Post columnist Gregg Dobbs two days earlier who wrote that he didn't believe Trump had the patience, intelligence or emotional equilibrium to deal with the rapidly escalating situation on the Korean Peninsula.   The basic reasons have been echoed many times from the date Trump "won" the farcical 2016 general election, by the likes of David Brooks, Charles Krauthammer, Charles Blow, Eugene Robinson, Richard Cohen and others.  Cohen early on even recommended Trump be removed under the little known 25th amendment - by virtue of his inability to rein in his egoistic petulance, along with his reckless bombast and narcissism. I agreed, and in a subsequent blog post asserted the Donald is barely able to run a dog kennel, far less an administration or the whole nation.

All these concerns have acquired enormous gravitas as the North Korean standoff has now reached the phase it could instantly turn into a blood bath.  With Trump now deploying the Carl Vinson carrier  strike group close to the peninsula along with the U.S.S. Michigan - a nuclear sub outfitted with nuclear missiles - we're now in a situation close the Cuban Missile Crisis of October, 1962. We got through that crisis because we had a man of temperance, intelligence and historical understanding at the helm : John Fitzgerald Kennedy..  Now, we have a "former" failed businessman who actually rode his sleazy real estate empire -  based largely on bankruptcies - to fame and fortune. Oh, after he already got 'nuff millions from daddy via inheritance.

So now we're saddled with a clown:  a bombastic pseudo-business tycoon and former reality TV personality who still thinks he's playing a role on 'The Apprentice' and believes any time he barks he can make lowly staff jump.  But we're in the real world now, not Trump's fantasy world of ego, entitlement and arrogance. And despite multiple warnings from even former military - generals and admirals - Trump can't seem to back off the bellicose rhetoric. This even as the North Koreans have vowed to take down the Vinson using one of their missiles. Rather than pause and reflect, Trump continues to think he's playing 'chicken' or high stakes poker - not that he's playing with the lives of tens of thousands.

As Gregg Dobbs put it, Kim Jong-un is not intimidated by threats, especially hollow ones - and also any pre-emptive strike would be about the worst move  possible. It would ensure the slaughter of tens of thousands of South Koreans as well as U.S. troops stationed there.   There is even the possibility the North Koreans could be spurred to employ one or more of their nuclear weapons, again most likely against South Korea - namely Seoul.

Things might not have reached this malignant stage had the media (most) not done back flips with wet dreams after Trump launched his 59 cruise missiles on a Syrian base. Since then, this deformed mental mutant has been putatively granted the "keys to the kingdom" (read: "respect" from media) by a stupid pack of slobbering suckers and goofballs who'd held their breath for any sign of Trump being "presidential".   Since these sycophantic imbeciles conflate use of force with "being presidential"  it was inevitable we'd now see a guy acting like he's been given his first hit of smack or crack cocaine.

As I wrote before, unless the same media pulls him back to reality - mainly by halting the cheerleading- we may now be in for the "most serious military confrontation since the Korean War" - in the words of one pundit last night.

Except this confrontation may be nuclear and we have a psychotic in charge of the country without the brains, knowledge, forbearance or basic human skills to deal with the situation.

Monday, April 24, 2017

Math Revisited: Looking At Linear Algebra

In earlier posts we looked at examples of linear algebra in terms of the behavior of lines and planes, e.g.

Now, we examine this fascinating branch of advanced math at deeper levels. Some aspects will resonate from when we were looked linear solutions of certain differential equations, i.e.

In this linear algebra context, let a 3 x 3 matrix A =

(a 1.....0.......0)
(0.......a 2.....0)
(0.......0.......a n)

We're first  interested in obtaining its characteristic polynomial from:

P_A(t) =

(t- a 1.....0.......0)
(0.......t – a 2.....0)
(0.......0....... t - a n)


P_A(t) = (t – a1)(t – a2) (t – a n)

The eigenvalues can be obtained via solving for a1, a 2, a n, in the equation:

(t – a1)(t – a2) (t – an) = 0


Given the matrix:

A =

(1.. ..i)

Find the characteristic polynomial as well as the eigenvalues.

We have:

P_A(t) =

(t - 1…… i)
( i....... t + 2)

Whence: P_A(t) = (t – 1)(t + 2) – (i)2  

P_A(t) = t 2 –t + 2t -2 - (i) 2 = t 2 +t -2 + 1= t2 + t – 1

Since this is a quadratic equation, so we can find the eigenvalues (E1,2) using the quadratic formula:

E1,2 = Ö{- b +  [b 2 – 4 ac]} / 2a

Where  a, b, c denote the coefficients for the quadratic, with a the numerical coefficient for the exponent 2 term (t2), b for the exponent 1 term(t) and c the exponent 0 term. Thus: a = 1, b = 1, c = -1

Then:  E1,2 = Ö{- 1 +/- [12 – 4 (-1)]} / 2(1)

E1,2= Ö{-1 +/- [5] } / 2

So that:

E1 = Ö(-1 + [5] ) / 2 = 0.618

E2=Ö (-1 - [5] ) / 2 = -1.618

Practice Problems:

Find the characteristic polynomials and eigenvalues for each of the following matrices:

 X =

(1.. …. .2)

 Y =

(3.. ……2)
(-2...... 3)