Matt Welch mentioned this earlier today, but it deserves extra attention and extra scorn: a "big group of foreign policy experts" -- that's what The Weekly Standard calls them, "foreign policy experts" -- urging "the United States and other willing nations" to "consider direct military strikes against the pillars of the Assad regime." The roster of "experts" is a sight to behold: Gary Bauer! Martin Peretz! Karl Rove! L. Paul Bremer!

I haven't written a lot about the possibly pending American intervention in Syria, because Jesus fucking Christ do people seriously want a war in Syria? To argue convincingly against an idea I need some capacity for understanding the other side of the argument, and at the moment my willingness to put myself in these jokers' shoes is pretty limited. Sorry. Dear experts: If you want to hear the most compelling case against your latest crusade, lock yourself in a room, strap yourself to a chair, and watch a rerun of the last 10 years. Maybe you'll learn something.

A conversation prompted by Simon’s last post reminded me of an analogy that’s too excellent to be buried in a comments thread. It must be very well-known, but I’ll go ahead and describe it anyway.

The analogy is between complex numbers and linear operators on an inner product space. Its best feature is that it makes important properties of complex numbers correspond to important properties of operators:

The title of this post refers to Sheldon Axler’s beautiful book Linear Algebra Done Right, which I’ve written about before. Most of what I’ll say can be found in Chapter 7. It’s one of those texts that feels like a piece of category theory even though it’s not actually about categories.

Today, all vector spaces are over ℂ and finite-dimensional. Most (all?) of what I’ll say can be done in more sophisticated functional-analytic settings, but I’ll stick to this most basic of situations.

Fix a vector space X equipped with an inner product. By an operator on X, I mean a linear map X→X.

Here’s how the analogy goes.

Complex numbers are like operators  This is the basis of everything that follows.

There’s not much substance to this statement yet. For now, let’s just observe that both the complex numbers and the operators on X form rings. I’ll write End(X) for the ring of operators on X, following the usual categorical custom. (“End” stands for “endomorphisms”.)

The two rings ℂ and End(X) don’t seem very similar. Unlike ℂ, the ring End(X) isn’t commutative and usually has nontrivial zero-divisors. (Indeed, as long as dim(X)≥2, there is some T∈End(X) with T≠0 but T2=0.) Perhaps surprisingly, these differences don’t prevent the development of this useful analogy.

In some loose sense, we can pass back and forth between ℂ and End(X). In one direction, starting with a complex number λ, we get the operator x↦λx. In elementary texts, this operator is often written as λI, but I’ll almost always write it as just λ.

In the opposite direction, starting with an operator on X, we get not just a single complex number but a collection of them — namely, its eigenvalues.

Complex conjugates are like adjoints  Every complex number z has a complex conjugate z*. Taking complex conjugates defines a self-inverse automorphism of the ring ℂ.

Every linear map T:X→Y of inner product spaces has an adjoint T*:Y→X, characterized by the equation ⟨Tx,y⟩=⟨x,T*y⟩. In particular, every operator T on X has an adjoint T*, also an operator on X.

It’s almost true that taking adjoints defines a self-inverse automorphism of End(X). The only obstruction is that taking adjoints reverses the order of composition: (TS)*=S*T*. So actually, taking adjoints defines a pair of mutually inverse ring isomorphisms

where End(X)op is the ring End(X) with its order of multiplication reversed.

What about those back-and-forth passages between complex numbers and operators?

First, start with a complex number λ; then the adjoint of the operator λI is λ*I. That is, (λI)*=λ*I. This is why I’m writing z* for the complex conjugate of z, rather than the more common z¯.

Second, start with an operator T. Then the eigenvalues of T* are exactly the conjugates of the eigenvalues of T. Why? Because taking the adjoint defines an isomorphism of rings, so T−λ is invertible iff (T−λ)*=T*−λ* is.

Real numbers are like self-adjoint operators  A complex number z is real if and only if z=z*. By definition, an operator T is self-adjoint if and only if T=T*.

Again, let’s look at the passages back and forth between ℂ and End(X). First, let λ∈ℂ. As long as X is nontrivial, the operator λ is self-adjoint iff λ is real.

Second, if T is a self-adjoint operator then all its eigenvalues are real. The converse isn’t true: an operator can have all real eigenvalues without being self-adjoint. We’ll come back to that.

Any even half-serious endeavour involving self-adjoint operators makes use of the theorem that classifies them, the spectral theorem. Loosely put, this states that every self-adjoint operator is an orthogonal sum of self-adjoint operators of the most simple kind: scalar multiplication by a real number.

Precisely: given any self-adjoint operator T, there is a unique orthogonal decomposition X=⨁λ∈ℝXλ such that for each λ, the restriction of T to Xλ is multiplication by λ. Of course, all but finitely many of these subspaces Xλ are trivial, the nontrivial ones are those for which λ is an eigenvalue, and Xλ is the eigenspace ker(T−λ).

Nonnegative real numbers are like positive operators  For a complex number z, the following are equivalent:

• (1) z is nonnegative, i.e. real and ≥0
• (2) z=w*w for some complex w
• (3) z=w*w (=w2) for some real w
• (4) z=w*w (=w2) for some nonnegative w
• (5) z=w*w (=w2) for a unique nonnegative w.

I’ll follow custom and say that an operator T is positive if it is self-adjoint and each eigenvalue is ≥0. (Other names are “positive semidefinite” and “nonnegative definite”. As we were recently discussing, the terminology around positive/nonnegative is a bit of a mess.) Note that by definition, “positive” includes “self-adjoint”. This is just like the convention that when we call a complex number “nonnegative”, we tacitly include the condition “real”.

For an operator T on X, the following are equivalent:

• (1) T is positive, i.e. self-adjoint and each eigenvalue is ≥0
• (1.5) T is self-adjoint and ⟨Tx,x⟩≥0 for all x∈X
• (2) T=S*S for some inner product space Y and linear map S:X→Y
• (2.5) T=S*S for some operator S on X
• (3) T=S*S (=S2) for some self-adjoint operator S
• (4) T=S*S (=S2) for some positive operator S
• (5) T=S*S (=S2) for a unique positive operator S.

The implications 5⇒4⇒⋯⇒1 are all either trivial or easy. The remaining implication, 1⇒5, follows from the spectral theorem, using 1⇒5 of the result on nonnegativity of numbers.

In particular, given λ∈ℂ, the operator λ is positive iff the number λ is nonnegative (assuming that X is nontrivial). And given an operator T, if T is positive then each eigenvalue of T is nonnegative (but not conversely).

The modulus of a complex number is like… the modulus of an operator? What is the modulus of a complex number? Let’s answer this carefully, using the theorem above on nonnegativity of complex numbers. Let z∈ℂ. By the theorem, z*z is nonnegative, so by the theorem again, there is a unique nonnegative m such that z*z=m*m (=m2). This m is, of course, ∣z∣, the modulus of z.

What is the analogue for operators? Let’s use the theorem above on positivity of operators. Let T∈End(X). By the theorem, T*T is positive, so by the theorem again, there is a unique positive M such that T*T=M*M (=M2). I’ll call M the modulus of T and write it as ∣T∣. I don’t know whether the term “modulus” is standard here, and I’m pretty sure the notation ∣T∣ isn’t — it’s risky, given the potential for confusion with a norm. But I’ll use it anyway, to emphasize the analogy.

Complex numbers of unit modulus are like isometries  A complex number z has unit modulus if and only if z*z=1, if and only if zz*=1. An operator T is an isometry if and only if T*T=1, if and only if TT*=1 (if and only if T preserves inner products, if and only if T preserves distances). Isometries are more often called unitary operators, but I find the term “isometry” more vivid.

Now that we have a definition of “modulus” for operators, we can ask: which operators are literally “of unit modulus”? In other words, which operators T satisfy ∣T∣=1? Here 1 is the identity operator. Certainly 1 is positive, so ∣T∣=1 if and only if T*T=1*1, if and only if T is an isometry. So the different parts of the analogy hang together nicely.

Once again, let’s go back and forth between complex numbers and operators. Given λ∈ℂ, the operator λ is an isometry iff the number λ is of unit modulus (again, assuming that X is nontrivial). Given an operator T, if T is an isometry then all its eigenvalues are of unit modulus. Again, the converse is false, and again, we’ll come back to that.

Polar decomposition of complex numbers and operators Any complex number z can be expressed as a product

where u is of unit modulus and p is nonnegative. Moreover, this p is uniquely determined as ∣z∣, and if z≠0 then u is uniquely determined by z too. (If z=0 then many choices of u are possible.)

Similarly, it’s a theorem that any operator T can be expressed as a composite

where U is an isometry and P is positive. Moreover, this P is uniquely determined as ∣T∣, and if T is invertible then U is uniquely determined by T too. (If T is not invertible then many choices of U are possible.)

In the case where T is just multiplication by a scalar z, the second theorem (polar decomposition of operators) reduces to the first (polar decomposition of complex numbers).

If you prefer, you can decompose an operator in the other order too: an isometry followed by a positive operator. To see this, decompose T* as UP; then T=P*U*=PU*. But U* is an isometry, since the adjoint of an isometry is again an isometry — just as the conjugate of a complex number of unit modulus is again of unit modulus.

And that’s the analogy.

Normal operators, and the fraying of the analogy

Like all analogies, this one eventually frays. Right at the start, we noted a big difference between complex numbers and operators: multiplying complex numbers is commutative, but composing operators isn’t. And another one: there are no nonzero nilpotent complex numbers, but there are nonzero nilpotent operators.

I’ll explain the trouble this causes by talking about operators T that satisfy the equation T*T=TT*. In a fit of no inspiration, someone once called such operators normal, and the name stuck.

Now, all complex numbers z are “normal”, in the sense that z*z=zz*, but not all operators T are normal — for example, any nonzero nilpotent is “abnormal”. So this is a wrinkle in the analogy. You might conclude from this that the correct analogue for the complex numbers is not the set of all operators, but just the normal ones. This idea has in its favour that all self-adjoint operators and isometries (“real numbers” and “numbers of unit modulus”) are normal — because an operator commutes with both itself and its inverse.

However, the normal operators don’t form a ring, at least, not under the usual operations. The class of normal operators is closed under taking polynomials in one variable, but not under composition. Indeed, the polar decomposition theorem implies that by composing two normal operators, we can obtain any operator we like.

The normal operators are nevertheless a useful class, giving further depth to the analogy. I clearly remember the first time I saw the definition of normal operator: I was overwhelmed by the feeling that it was an awful hack. “Someone,” I thought to myself, “simply wants a definition that includes both self-adjoint operators and isometries, and they’ve written down the first thing that came into their head.” Oh young, foolish self; I was wrong. Here’s why:

Normal operators are exactly the right context for the spectral theorem.

Recall that for an operator T, the spectral theorem says that X is the orthogonal sum of the eigenspaces of T. This statement isn’t true for all operators. Earlier on, I stated that it was true for all self-adjoint operators, and that in that case, all the eigenvalues are real. But there are certainly non-self-adjoint operators such that X is the orthogonal sum of the eigenspaces — multiplication by any non-real scalar is an example.

So which operators is the spectral theorem true for? Exactly the normal ones. In other words:

Spectral theorem  Let T be an operator on X. Then X is the orthogonal sum of the eigenspaces of T if and only if T is normal.

This says that multiplication by a scalar is a normal operator, that the class of normal operators is closed under orthogonal sums, and that combining these two constructions generates all possible normal operators. ‘Only if’ is easy; it’s ‘if’ that takes work. You can find a proof in Linear Algebra Done Right.

We can read off two corollaries, both supporting the claim that “complex numbers are like normal operators” is a better analogy than “complex numbers are like operators”.

Corollary  Let T be a normal operator. Then (i) T is self-adjoint if and only if all eigenvalues of T are real, and (ii) T is an isometry if and only if all eigenvalues of T are of unit modulus.

We saw earlier that without the normality, the “only if” parts are true but the “if” parts fail.

Fundamental theorem of algebra for normal operators  Let p be a nonconstant polynomial over ℂ, and let T be a normal operator. Then there exists a normal operator S such that p(S)=T.

For both proofs, all we have to do is observe that the class of operators T for which the result holds contains all operators of the form “multiply by a scalar” and is closed under orthogonal sums. That’s all there is to it!

Like the literary pirate captain from whom he borrowed his name, Silk Road proprietor Dread Pirate Roberts is the successor to the actual founder of a criminal enterprise — although, in the real world case, it's the victimless activity of peddling forbidden intoxicants and other illicit goods to willing buyers. As fascinating as the encrypted and anonymous online black market is, though, it's made even more intriguing by the Dread Pirate Roberts's libertarian philosophical musings. 

From Andy Greenberg at Forbes:

Roberts also has a political agenda: He sees himself not just as an enabler of street-corner pushers but also as a radical libertarian revolutionary carving out an anarchic digital space beyond the reach of the taxation and regulatory powers of the state–Julian Assange with a hypodermic needle. “We can’t stay silent forever. We have an important message, and the time is ripe for the world to hear it,” says Roberts. “What we’re doing isn’t about scoring drugs or ‘sticking it to the man.’ It’s about standing up for our rights as human beings and refusing to submit when we’ve done no wrong.”

“Silk Road is a vehicle for that message,” he writes to me from somewhere in the Internet’s encrypted void. “All else is secondary.” ...

“We’re talking about the potential for a monumental shift in the power structure of the world,” Roberts writes. “The people now can control the flow and distribution of information and the flow of money. Sector by sector the State is being cut out of the equation and power is being returned to the individual.”

Roberts's ideas are pretty specific. Greenberg writes that "he’s even hosted a Dread Pirate Roberts Book Club where he moderated discussions on authors from the Austrian school of free market economics."

You don't need ideology to participate in a successful underground business, however, and intriguing ideas won't make such a venture fly. An earlier interview by Vice with some of the dealers who sell through Silk Road found that they were "really nice guys" who were very concerned about customer service (Silk Road has seller ratings and holds payments in escrow until goods are delivered).

The technology on which Silk Road and Roberts rely — Tor and Bitcoin — are nominally neutral, but inherently political, since they allow for free and anonymous transactions with or without the consent of the state. That certainly explains why so many government officials are openly hostile to both encryption and digital currencies. And, then again, the control-driven antagonism to such technologies is exactly what drives their development

Says Roberts in the extended interview:

At its core, Silk Road is a way to get around regulation from the state. If they say we can’t buy and sell certain things, we’ll do it anyway and suffer no abuse from them. But the state tries to control nearly every aspect of our lives, not just drug use. Anywhere they do that, there is an opportunity to live your life as you see fit despite their efforts.

Next up, suggests Roberts, is a renewed effort to sell firearms and ammunition online, to escape tightened controls around the world. Also, the site is looking at basic consumer electronics, since high tariffs have created an opening for black market operators.

As Reason's Matthew Feeney noted, Silk Road's success has spawned competition, most notably the recent startup, Atlantis.

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Via the Guardian, for whom Glenn Greenwald has done much reporting, including a lot of the best stuff on Edward Snowden's NSA revelations:

The partner of the Guardian journalist who has written a series of stories revealing mass surveillance programmes by the US National SecurityAgency was held for almost nine hours on Sunday by UK authorities as he passed through London's Heathrow airport on his way home to Rio de Janeiro.

David Miranda, who lives with Glenn Greenwald, was returning from a trip to Berlin when he was stopped by officers at 8.30am and informed that he was to be questioned under schedule 7 of the Terrorism Act 2000. The controversial law, which applies only at airports, ports and border areas, allows officers to stop, search, question and detain individuals.

The 28-year-old was held for nine hours, the maximum the law allows before officers must release or formally arrest the individual. Accordingto official figures, most examinations under schedule 7 – over 97% – last under an hour, and only one in 2,000 people detained are kept for more than six hours.

Miranda was then released without charge, but officials confiscated electronics equipment including his mobile phone, laptop, camera, memory sticks, DVDs and games consoles...

Reason on Greenwald.

UPDATE: Greenwald on his experience:

At 6:30 am this morning my time - 5:30 am on the East Coast of the US - I received a telephone call from someone who identified himself as a "security official at Heathrow airport." He told me that my partner, David Miranda, had been "detained" at the London airport "under Schedule 7 of the Terrorism Act of 2000."....

At the time the "security official" called me, David had been detained for 3 hours....The official - who refused to give his name but would only identify himself by his number: 203654 - said David was not allowed to have a lawyer present, nor would they allow me to talk to him.

I immediately contacted the Guardian, which sent lawyers to the airport, as well various Brazilian officials I know. Within the hour, several senior Brazilian officials were engaged and expressing indignation over what was being done....

Despite all that, five more hours went by and neither the Guardian's lawyers nor Brazilian officials, including the Ambassador to the UK in London, were able to obtain any information about David....

According to a document published by the UK government about Schedule 7 of the Terrorism Act, "fewer than 3 people in every 10,000 are examined as they pass through UK borders" (David was not entering the UK but only transiting through to Rio). Moreover, "most examinations, over 97%, last under an hour." An appendix to that document states that only .06% of all people detained are kept for more than 6 hours.

The stated purpose of this law, as the name suggests, is to question people about terrorism....

But they obviously had zero suspicion that David was associated with a terrorist organization or involved in any terrorist plot. Instead, they spent their time interrogating him about the NSA reporting which Laura Poitras, the Guardian and I are doing, as well the content of the electronic products he was carrying. They completely abused their own terrorism law for reasons having nothing whatsoever to do with terrorism: a potent reminder of how often governments lie when they claim that they need powers to stop "the terrorists", and how dangerous it is to vest unchecked power with political officials in its name...

UPDATE II: According to the New York Times, the targeting of Miranda--while still having nothing whatever to do with terrorism investigations, remember--was not random harassment either. They apparently knew that Miranda had Snowden-related documents on a thumb drive on his person, which they stole, ones that documentary filmmaker Laura Poitras had given him to bring to Greenwald.

The stunning Danxia Scenic Area in Zhangye City, northwest China's Gansu Province. Danxia, which means rosy cloud, is a special landform formed from reddish sandstone that has been eroded over time into a series of mountains surrounded by curvaceous cliffs and many unusual rock formations. Picture: CATERS

I wrote a piece for The Washington Post about political paranoia and the war on leaks. Here's the opening:

In the popular stereotype, conspiracy theorists direct their paranoia at the government: The CIA shot JFK. NASA faked the moon landing. Sept. 11 was an inside job.

But the most significant sorts of political paranoia are the kinds that catch on with people inside the halls of power, not the folks on the outside looking in. The latest example is a crackdown on leaks that has the government crippled by a fear of its own employees. Washington is petrified of itself.

The federal effort, called the Insider Threat Program, was launched in October 2011, and it certainly hasn’t diminished since Edward Snowden disclosed details of the National Security Agency’s domestic spying. As McClatchy reporters Marisa Taylor and Jonathan S. Landay have described, federal employees and contractors are encouraged to keep an eye on allegedly suspicious “indicators” in their co-workers’ lives, from financial troubles to divorce. A brochure produced by the Defense Security Service, titled “INSIDER THREATS: Combating the ENEMY within your organization,” sums up the spirit of the program: “It is better to have reported overzealously than never to have reported at all.”

To read the rest, in which I compare the Leak Scare to earlier fears, go here.

In a July 13, 2013 memo unearthed by the group Consumer Watchdog, Google told the U.S. District Court for Northern District of California this about their email customers:

Indeed, “a person has no legitimate expectation of privacy in information he voluntarily turns over to third parties.” Smith v. Maryland, 442 U.S. 735, 743-44 (1979). In particular, the Court noted that persons communicating through a service provided by an intermediary (in the Smith case, a telephone call routed through a telephone company) must necessarily expect that the communication will be subject to the intermediary’s systems. For example, the Court explained that in using the telephone, a person “voluntarily convey[s] numerical information to the telephone company and ‘expose[s]’ that information to its equipment in the ordinary course of business.”

That looks like bad news for Gmail users like me, and seems particularly sinister as we try to untangle how complicit various companies have been in the NSA surveillance dustup. 

But unfortunately for all of us, this problem isn't confined to Google—the company was simply citing Supreme Court precedent in that memo. The Court's interpretation of the Fourth Amendment has swung so far away from the kind of privacy protections most Americas vaguely believe they enjoy that Google is correct, there is no "expectation of privacy." 

For more on this point, watch Matt Welch interview Judge Alex Kozinski, queued up to an interesting take on privacy and the Fourth Amendment below:

Business Insider digs up Google Executive Chairman Eric Schmidt, sounding like a bit of jerk on CNBC in 2009 on a related point: 

"If you have something that you don't want anyone to know, maybe you shouldn't be doing it in the first place," Schmidt said. "But if you really need that kind of privacy, the reality is that search engines, including Google, do retain this information for some time. And ... we're all subject, in the United States, to the Patriot Act, and it is possible that that information could be made available to the authorities."

Google says its scanning your emails to provide services people want and also sell them stuff. The government is using that same data to look for reasons to put you in jail.

Two email services that purported to provide more privacy, Lavabit and Silent Circle, recently shut down, citing federal investigations, which further suggests that the problem isn't Google alone, but the general legal climate.

Here's the whole motion:

Bill McMorris covered this year's Gathering of the Juuggalos -- the annual convocation of Insane Clown Posse fans -- for The Washington Free Beacon. His dispatch includes this moment of legal anthropology, which I submit without comment:

Juggalo Night Court [is] a daily ceremony designed to smooth out trivial festival disputes under the jurisprudence of Judge Upchuck the Clown and bailiff/professional wrestler Mad Man Pondo.

I took my seat on the haystacks on Wednesday night. The matter at hand: Juggalo Lee had sued Juggalo Pete for groping his date.

"It was the heat of the moment," Pete said before admitting to groping at least a dozen other women. During closing arguments, the alleged victim took the stage topless and allowed Pondo and Upchuck to grope her.

The crowd sided with Pete. Lee was tarred and feathered with honey and a gutted pillow.

Lust is forgivable in Juggalo eyes. Theft is not. The Gathering program warns of "dire consequences" for stealing, and that's an understatement. Last year a man was found with pilfered goods in the trunk of his red Pontiac. Juggalos stripped his car, smashed the windshield, ran it over with a monster truck, then posted the video to YouTube under the title "Juggalo Justice."

Juggalo Justice is why I feel safe leaving my laptop in public, my car doors open, and beers unattended.

Bonus quote for conspiracy buffs: "Juggalos see themselves under constant threat—every one swears to the existence of Juggalo Holocaust, a mythical entity hell-bent on killing ICP fans." Of course, the Juggalos themselves are perceived as a conspiracy in some quarters.

Florida's famously dumb legislature just one-upped itself by banning all computers and smart phones, according to a lawsuite filed by the owner of an internet cafe "whose clientele is primarily migrant workers seeking computer time."

Consuelo Zapata says in her suit that a law passed in the wake of a gambling scandal involving Lt. Gov. Jennifer Carroll doesn't just forbid illegal computer gambling and slot machines, but all computers. Mary Ellen Klas of the Tampa Bay Times reports: 

The lawsuit filed in Miami-Dade Circuit Court on behalf of Incredible Investments, LLC, owned by Consuelo Zapata, alleges that the Legislature effectively applied the ban to all computers when it defined illegal slot machines as any "system or network of devices" that may be used in a game of chance. The state effectively made every smartphone and computer an illegal device, the plaintiff argues.

"They rushed to judgment and they took what they saw as a very specific problem and essentially criminalized everything," said Justin Kaplan of the Miami law firm of Kluger, Kaplan, Silverman, Katzen & Levine, which is representing Zapata.

The argument, crafted with the help of constitutional law attorney and Harvard professor Alan Dershowitz, is based on the assumption that the definition of illegal slot machines is now so broad that an illegal game could be potentially played on every computer. Under the law, the Legislature's own computers, "the ones they used to draft this legislation, are illegal,'' Kaplan said.

The case asks the court to throw out the law passed "in a frenzy fueled by distorted judgment in the wake of a scandal that included the lieutenant governor's resignation." It argues that the lawsuit unlawfully prohibits, commerce, violates free speech and due process and is overly broad and unworkably vague.

I also take this to mean that it's illegal to have an office intranet. 

Previously in Florida legislative stupidity: Florida Just Made It a Crime to Sell Bongs, Glass Pipes, Etc.