An Introduction to Linear Preserver Problems

August 5th, 2010

The theory of linear preserver problems deals with characterizing linear (complex) matrix-valued maps that preserve certain properties of the matrices they act on. For example, some of the most famous linear preserver problems ask what a map must look like if it preserves invertibility or the determinant of matrices. Today I will focus on introducing some of the basic linear preserver problems that got the field off the ground – in the near future I will explore linear preserver problems dealing with various families of norms and linear preserver problems that are actively used today in quantum information theory. In the meantime, the interested reader can find a more thorough introduction to common linear preserver problems in [1,2].

Suppose Φ : Mn → Mn (where Mn is the set of n×n complex matrices) is a linear map. It is well-known that any such map can be written in the form

where {Ai}, {Bi} ⊂ Mn are families of matrices (sometimes referred to as the left and right generalized Choi-Kraus operators of Φ (phew!)). But what if we make the additional restrictions that Φ is an invertible map and Φ(X) is nonsingular whenever X ∈ Mn is nonsingular? The problem of characterizing maps of this type (which are sometimes called invertibility-preserving maps) is one of the first linear preserver problems that was solved, and it turns out that if Φ is invertibility-preserving then either Φ or T ○ Φ (where T represents the matrix transpose map) can be written with just a single pair of Choi-Kraus operators:

Theorem 1. [3] Let Φ : Mn → Mn be an invertible linear map. Then Φ(X) is nonsingular whenever X ∈ Mn is nonsingular if and only if there exist M, N ∈ Mn with det(MN) ≠ 0 such that

In addition to being interesting in its own right, Theorem 1 serves as a starting point that allows for the simple derivation of several related results.

Determinant-Preserving Maps

For example, suppose Φ is a linear map such that det(Φ(X)) = det(X) for all X ∈ Mn. We will now find the form that maps of this type (called determinant-preserving maps) have using Theorem 1. In order to use Theorem 1 though, we must first show that Φ is invertible.

We prove that Φ is invertible by contradiction. Suppose there exists X ≠ 0 such that Φ(X) = 0. Then because Φ preserves determinants, it must be the case that X is singular. Then there exists a singular Y ∈ Mn such that X + Y is nonsingular. It follows that 0 ≠ det(X + Y) = det(Φ(X + Y)) = det(0 + Φ(Y)) = det(Y) = 0, a contradiction. Thus it must be the case that X = 0 and so Φ is invertible.

Furthermore, any map that preserves determinants must preserve the set of nonsingular matrices because X is nonsingular if and only if det(X) ≠ 0. It follows from Theorem 1 that for any determinant-preserving map Φ there must exist M, N ∈ Mn with det(MN) ≠ 0 such that either Φ(X) = MXN or Φ(X) = MXTN. However, in this case we have det(X) = det(Φ(X)) = det(MXN) = det(MN)det(X) for all X ∈ Mn, so det(MN) = 1. Conversely, it is not difficult (an exercise left to the interested reader) to show that any map of this form with det(MN) = 1 must be determinant-preserving. What we have proved is the following result, originally due to Frobenius [4]:

Theorem 2. Let Φ : Mn → Mn be a linear map. Then det(Φ(X)) = det(X) for all X ∈ Mn if and only if there exist M, N ∈ Mn with det(MN) = 1 such that

Spectrum-Preserving Maps

The final linear preserver problem that we will consider right now is the problem of characterizing linear maps Φ such that the eigenvalues (counting multiplicities) of Φ(X) are the same as the eigenvalues of X for all X ∈ Mn (such maps are sometimes called spectrum-preserving maps). Certainly any map that is spectrum-preserving must also be determinant-preserving (since the determinant of a matrix is just the product of its eigenvalues), so by Theorem 2 there exist M, N ∈ Mn with det(MN) = 1 such that either Φ(X) = MXN or Φ(X) = MXTN.

Now note that any map that preserves eigenvalues must also preserve trace (since the trace is just the sum of the matrix’s eigenvalues) and so we have Tr(X) = Tr(Φ(X)) = Tr(MXN) = Tr(NMX) for all X ∈ Mn. This implies that Tr((I – NM)X) = 0 for all X ∈ Mn, so we have NM = I (i.e., M = N-1). Conversely, it is simple (another exercise left for the interested reader) to show that any map of this form with M = N-1 must be spectrum-preserving. What we have proved is the following characterization of maps that preserve eigenvalues:

Theorem 3. Let Φ : Mn → Mn be a linear map. Then Φ is spectrum-preserving if and only if det(Φ(X)) = det(X) and Tr(Φ(X)) = Tr(X) for all X ∈ Mn if and only if there exists a nonsingular N ∈ Mn such that

References:

  1. C. K. Li, S. Pierce, Linear preserver problems. The American Mathematical Monthly 108, 591–605 (2001).
  2. C. K. Li, N. K. Tsing, Linear preserver problems: A brief introduction and some special techniques. Linear Algebra and its Applications 162164, 217–235 (1992).
  3. J. Dieudonne, Sur une generalisation du groupe orthogonal a quatre variables. Arch. Math. 1,
    282–287 (1949).
  4. G. Frobenius, Uber die Darstellung der endlichen Gruppen durch Linear Substitutionen. Sitzungsber
    Deutsch. Akad. Wiss. Berlin 994–1015 (1897).

Lifeline is Now Online

March 22nd, 2010

Lifeline was a newsletter for “enthusiasts of Conway’s Game of Life” that was published by Robert Wainwright in the early 1970′s. It and a handful of Scientific American articles were some of the only places ever to describe and coordinate the multitude of discoveries in the game during its early years. It ran for 11 issues from March 1971 through September 1973 and it was the first place in which the following discoveries/inventions (among others) were described:

Unfortunately, Lifeline has been extremely difficult to find because only about 500 copies of the newsletter were distributed and they were distributed some 40 years ago. Thanks to someone who found a set of the newsletters in the bottom of a box somewhere however, I have been able to scan them and get all eleven issues online at the LifeWiki:

Lifeline Number 7

Number 1Number 2Number 3Number 4Number 5Number 6

Number 7Number 8Number 9Number 10Number 11

All eleven issues have page scans provided as images, the first five issues have been transcribed to text, and the first four issues have had their images updated/pretty-ified a bit. Also, you can download a PDF of the first issue below. So go read and learn about the early days of Life! And if you’re feeling generous, help transcribe some of the later issues to text.

Download: Lifeline Number 1 [pdf — 5.52MB]

31,192-generation methuselah found in Conway's Game of Life

January 15th, 2010

One of the more exciting things to happen in my world this week was the discovery of a methuselah with lifespan 31,192 in Conway’s Game of Life. A methuselah is a small pattern that behaves chaotically for a large number of generations before settling down into a predictable mess.

The pattern, which has been named “Edna” (after Methuselah’s wife), was found by Erik de Neve via the Online Life-Like CA Soup Search and is the second notable discovery of the soup search (the first being the first infinitely-growing pattern in the 2×2 rule). Edna is now the longest-lived known (non-infinitely-growing) pattern that fits within a 20×20 square, beating the previous record-holder by over 2,000 generations.

Congratulations to Erik for finding the pattern after evolving a whopping 425 million random patterns.

The original 31,192-generation form of Edna

The original 31,192-generation form of Edna

A 26-cell form of Edna with lifespan 31,082

A 26-cell form of Edna with lifespan 31,082

Nerd Culture Calculus Worksheets

December 25th, 2009

I presented the labs for the first-year calculus course at my school this past semester, and as a bit of an experiment I decided to try giving the students some less “ordinary” problems to work on at the end of the labs (partly inspired by this problem). I only ended up doing it for the first four weeks of the semester due to a combination of it taking much too long to create them and a general lack of interest by most of the students, but they were fun to make anyway so I might as well share them in case anyone else would like to present these or similar problems in their own labs or course. PDF as well as TeX files are provided, so you can edit out my name and all that jazz.

Lab #1: Intervals with Braid

The first week’s problem was based off the video game Braid. This problem ended up not working too well due to about 10 people in the class of ~600 having played of the game, and the rest being very confused by the idea of the cloud platform moving along with the main character (it makes sense if you’ve played the game, honest!).

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Intervals with Braid

Lab #2: The Bat Man

The next week I decided to go a bit more mainstream and have the problem based on Batman chasing the Joker. The question doesn’t make a lick of sense if you think about it physically (the cars have negative acceleration for one thing), but this being a math class I decided not to care. I feel like this was the most successful of the weekly problems because the Batman/Joker stuff was completely incidental and the question was still easy enough for the students to understand and tackle.

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

The Bat Man

Lab #3: Calvin Reaches his Limit

By this point I had learned that even if I am going to sugar-coat the question under a picture of Calvin and Hobbes, it’s a good idea to include a sentence at the end summarizing what the heck it is I’m asking them to compute or prove. I think this is a fun question no matter how advanced of a mathematician you are, and it was probably a bit mean of me to present it to first-year students.

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Calvin Reaches his LimitLab #4: Continuity with Mario

The last of these problems that I presented was a (very) simple continuity question based on Super Mario World. I originally wanted to use Super Metroid for this question since it would allow for more varied movements from the hero, but I decided that (as I learned in Lab #1) it would be best to stick with a more recognizable game. It was a pain to come up with a semi-nice-looking branch function that resembled Mario’s movement in a believable way and led to simple (i.e., non-fractional) limits at the points of interest.

Download: Question sheet [pdf], Solution sheet [pdf], TeX files [zip]

Continuity with Mario

The Other Superoperator Isomorphism

November 20th, 2009

A few months ago, I spent two posts describing the Choi-Jamiolkowski isomorphism between linear operators from Mn to Mm (often referred to as “superoperators“) and linear operators living in the space Mn ⊗ Mm. However, there is another isomorphism between superoperators and regular operators — one that I’m not sure of any name for but which has just as many interesting properties.

Recall from Section 1 of this post that any superoperator Φ can be written as

\Phi(X)=\sum_iA_iXB_i.for some operators {Ai} and {Bi}. The isomorphism that I am going to focus on in this post is the one given by associating Φ with the operator

M_\Phi:=\sum_iA_i\otimes B_i^{T}.

The main reason that MΦ can be so useful is that it retains the operator structure of Φ. In particular, if you define vec(X) to be the vectorization of the operator X, then

{\rm vec}(\Phi(X))=M_\Phi{\rm vec}(X).

In other words, if you treat X as a vector, then MΦ is the operator describing the action of Φ on X. From this it becomes simple to compute some basic quantities describing Φ. For example, the induced Frobenius norm,

\big\|\Phi\big\|_F:=\sup_{\|X\|_F=1}\Big\{\big\|\Phi(X)\big\|_F\Big\},

is equal to the standard operator norm of MΦ. If n = m then we can define the eigenvalues {λ} and the eigenmatrices {V} of Φ in the obvious way via

\Phi(V)=\lambda V.

Then the eigenvalues of Φ are exactly the eigenvalues of MΦ, and the corresponding eigenvectors of MΦ are the vectorizations of the eigenmatrices of Φ. It is similarly easy to check whether Φ is invertible (by checking whether or not det(MΦ) = 0), find the inverse if it exists, or find the nullspace (and a pseudoinverse) if it doesn’t.

Finally, here’s a question for the interested reader to think about: why is the transpose required on the Bi operators for this isomorphism to make sense? That is, why can we not define an isomorphism between Φ and the operator

\sum_iA_i\otimes B_i?

Keep the "Info" Before the "Graphic"

November 13th, 2009

The term “infographic” is a ridiculous little buzzword that really took off on the internet sometime last year. It used to refer to genuinely useful things like subway maps and blueprints. Recently, however, the term has come to mean “an obnoxiously oversized image that has numbers on it”. My problem isn’t with infographics like these ones that just display some fun, meaningless information is a visual way, or this one that displays a phenomenon that is inherently visual. My beef is with infographics that reduce a variety of related statistics to an oversized mess of overlapping graphs and charts that are (purposely or otherwise) misleading.

This post will present four rules that infographic designers, if they decide that they absolutely must make an infographic, should always follow (but often don’t). To get the ball rolling, let’s consider an example that made its way around the internet just a couple of weeks ago (source):

american-2009-season-ratings

American 2009 Season Premieres and Averages to Date (click to enlarge)

We are told that the above infographic depicts the US viewership for a variety of shows during their premiere (light red) and on average since they began their 2009 season (dark red). However, I have two main problems with the image, and they’re both problems that are prevalent throughout many infographics and can easily be solved by just using a simple bar graph.

1. Infographics should not require horizontal scrolling. The above infographic is 3133 pixels wide, which means there is no consumer-available monitor in the world capable of displaying the entire image on one screen without scrunching it down. This is apparently exactly what infographic makers want, since they all seem to subscribe to the school of thought that dictates their image deserves 45 inches of horizontal viewing space. This would be fine if infographics were readable when zoomed out, but by their very nature they almost never are.

Computer monitors were not meant to view posters. If you want to make the image high-resolution enough that it can be printed out as a poster then it should be created as a vector graphic, not a raster graphic. If you still insist that your infographic should be a monstrously large bitmap, make it readable from a zoom level that will fit on standard monitor resolutions.

Some other popular infographics that suffer from this problem are the new auto industry breakdown, weight of the world, and the first 100 days.

2. Two-dimensional figures should never be used to compare linear data. The above infographic compares the number of people watching different shows, so why are circles being used to represent the data? What represents the number of viewers — the radius of the circle or the area of the circle? The source doesn’t tell us, so we have no way of appropriately assessing how many more people are viewing NCIS: Los Angeles than The Good Wife. If it’s the radius of the circle, NCIS appears to have about 5% more viewers. If it’s the area of the circle then it’s probably over 10% (and the discrepancy gets much larger if you compare shows that are farther apart).

Furthermore, even if we were told whether it’s the radii of the circles or their areas that we should be looking at, there’s still a problem. If the radii are what are being compared, then the visual is misleading because the differences in areas cause the relative differences to appear larger than they actually are. If the areas are what are being compared, then it should be noted that people just plain suck at visually comparing areas. By looking at the above image (and not getting out a ruler or anything) can you tell which circles have about half as much area as the NCIS: Los Angeles circle? Can you tell how much higher the viewership of The Good Wife is than that of Glee? I certainly can’t, at least not quickly.

InfomationIsBeautiful.net is a particularly notorious violator of this rule, as these three examples show: deadliest drugs, how safe is the HPV vaccine?, reduce your chances of dying in a plane crash (scroll down to the “bad month” and “the odds” sections). What’s worse is they aren’t even consistent with whether it’s the areas of the circles or the radii of the circles they’re comparing.

Problems #1 and #2 can both be rectified by simply turning the data into a bar graph. A plain old-fashioned bar graph. Voila:

American 2009 Season Premieres and Averages to Date (easier to read)

American 2009 Season Premieres and Averages to Date (easier to read)

The above bar graph doesn’t need to be zoomed in to be read, it makes it easier to compare the relative viewership of each show, and it actually contains more data than the previous infographic thanks to the labels on the vertical axis.

The next example (source) supposedly explains how and why low-cost airlines are able to offer flights that are so much cheaper than other airlines. It made its rounds this last spring during recession fever, when anything that had anything to do with something being cheap was instantly popular. While it does not suffer from problem #1 above (since it is readable when zoomed out), it suffers from two instances of problem #2 as well as multiple other problems.

How come airlines are so cheap?

How come cheap airlines are so cheap? (click to enlarge)

3. Infographics (and everything else) should be about substance over style. While there’s no denying that the above infographic is pretty, does it actually tell us anything? Beyond the myriad of small problems such as the average fare of Southwest flights including cents when none of the other numbers do, the misspelling of “Aer Lingus” and “maintenance”, and the mysterious 43% “total advantage” at the bottom that seems to pop out of nowhere, the infographic at its core doesn’t even make sense.

As the infographic itself says, low-cost airlines generally don’t do long-haul flights; they focus on short point-to-point routes. So why are their average fares being compared to the average fares of the likes of British Airways, who regularly do intercontinental flights? Doesn’t it make sense that travel distance makes more of a contribution to the price of the flight than whether or not tickets are sold primarily online? Average fare per kilometer travelled would make more sense to compare, though it would still be misleading because take-off and landing are disproportionately expensive.

Another recent offending infographic that just simply doesn’t say a thing is the $400 million club, which notes that Transformers: Revenge of the Fallen is only the ninth movie in history to gross more than $400 million at the box office in the US during its theatrical run. The infographic then compares the other eight movies, which of course are juggernauts like Star Wars and Titanic. The problem is that none of the figures are adjusted for inflation. If you scale the numbers properly, Transformers: Revenge of the Fallen actually comes in as about the 65th highest-grossing movie. Impressive, sure, but to say that the infographic is misleading is an understatement.

I will finish by presenting a graphic that ran on NewsWeek.com that shows obesity and “life evaluation” trends over the last year or two. It’s debatable whether or not it falls into the category of what most people would consider an “infographic”, but it perfectly illustrates a core problem with them.

Obesity infographic

4. Be careful with your data. Just making your graphic pretty doesn’t give you free reign to ignore basic statistical principles when presenting data. In the above graphic, the left graph shows two lines — one showing how many people have BMI less than or equal to 30 in a given month and one showing how many people have BMI over 30 in a given month. I have a news flash for you, NewsWeek: one of those lines is redundant. Not only that, but the redundant second line manipulates the reader by giving the false impression that the number of obese people is converging toward the number of non-obese people. Nevermind the fact that the vertical scale is completely out of whack and it jumps a vertical distance of 46.4% in the same amount of space that is used to represent about a 2.5% jump elsewhere.

I’m willing to bet that the vertical scale on the right graph is completely out of whack too, but it’s a little difficult to tell since they don’t tell you what percentages any of the intermediate y-values correspond to. On the blue “struggling” line, we are given a value of 48.4% on the left edge of the graph and a value of 49.6% at the right edge of the graph at a nearly identical height. Are we supposed to be able to tell how high and low the peaks in the middle of the graph are based on that? Does the blue line get as low as 40%? 35%? 30%? Would labels along the vertical axis (similar to the bar graph I showed above) really have detracted from the desired aesthetic too much?

So if you have a set of data that you wish to convey graphically, please first consider whether or not it can be presented by a simple bar graph or line graph. If it can, don’t try to make it more complicated than that. If it can’t, at least make sure that the information is the motivating factor in your decisions. If the layout ends up dictating how you present your data, you’ve got your priorities backward.

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