Of all the gaps matriculating from Circle Pines U left in my psyche -- how to knot a cravat, the difference between good flan and bad flan, world history before the 17th century or US after the 19th -- failure to even mention Gershgorin circles stands as a particularly egregious transgression.
In sophomore year we deconstructed Descartes' Meditations, this in the guise of a philosophy graduate student opining on the tangible gustatory effects of imagined peanut butter sandwiches. Perhaps it was a metaphor for sexual arousal, which would have made for more spirited recitation albeit risking pointed inquiry from the CP Chamber of Commerce, who already did not much care for our fancy book lernin'. Alas, whatever his intent, it was lost on us.
In junior year we discussed Lucifer State, a novel of dystopian future I rather enjoyed but is no longer in print, or so the Internet tells me. (Such not being the damning praise it once was, what with the Amazon self-publication machine Heutzutage delivering any provocatively-titled finger barfing -- Moist Vibrations, Debbie Does Due Diligence, Triggered -- to the tenterhooked browser.) Mostly I remember the prof lectured wearing sunglasses, which on more than one occasion concealed a black eye.
Then in senior year there was Herman Melville and the Forms, which was a comparative literature course. Unless that was freshman year, then it was a punk band. I think. TBH, the whole experience is something of a blur. I took classes in economics and FORTRAN, racquetball and contract law, geology and differential equations. I'm not entirely sure what my major was.
Someone could have devoted a lecture to Gershgorin circles, is my point. As I hope I have convinced you, the remainder of my education would not have suffered.
Besides, Gershgorin circles are muy cool.
Here's the deal.
Take any nxn real matrix. You are now going to draw n circles on the real line. Each circle is centered at the one of the diagonal entries of the matrix and the circle radius is equal to the sum of the absolute values of the off-diagonal terms in its row.
Believe it or not, all of the eigenvalues of the matrix live in those circles.
Wait, there's more!
Play the game with the columns instead of the rows. All of the eigenvalues live in that collection of circles, too.
Wait there's more!
Take the intersection of the row circles and the column circles. All of the eigenvalues live in that collection of circles, too.
The game also works if the matrix entries are complex. Now, however, you need to draw your circles in the complex plane rather than on the real line.
All this seems too splendiferous to be true, so I made a Javascript that demonstrates the 2x2 case:
Enter matrix entries* (or click the "random matrix" button). The app draws two circles and lists the two eigenvalues. Convince your noggin the latter live in the former. (NB: The x-axis is the real axis, but the plane is the complex plane.)
* There's minimal error checking in the code, and by "minimal" I mean none. So don't get sassy with your input.
These are known as Gershgorin circles, after S.A. Gershgorin who described them in 1931. A footnote in Meyer says the idea appeared in previous work by Lévy, Minkowski, and Hadamard, but mathematicians Olga Taussky and Alfred Brauer worked to get Gershgorin's name attached to it because reasons (Lévy, Minkowsky, and Hadamard got other stuff named after them).
In the maelstrom of agony that is mathematics on any given day, such a simple something is a wild gift. A cosmic thing. Useful, too. For example, Gershgorin immediately tells you all diagonally dominant matrices are nonsingular (think about it). Other delights and connections await discovery, should you care to explore.
A proof the circles work as advertised is not terribly complicated, but I'll let you suss it out or look it up yourself.
Gershgorin!
In sophomore year we deconstructed Descartes' Meditations, this in the guise of a philosophy graduate student opining on the tangible gustatory effects of imagined peanut butter sandwiches. Perhaps it was a metaphor for sexual arousal, which would have made for more spirited recitation albeit risking pointed inquiry from the CP Chamber of Commerce, who already did not much care for our fancy book lernin'. Alas, whatever his intent, it was lost on us.
In junior year we discussed Lucifer State, a novel of dystopian future I rather enjoyed but is no longer in print, or so the Internet tells me. (Such not being the damning praise it once was, what with the Amazon self-publication machine Heutzutage delivering any provocatively-titled finger barfing -- Moist Vibrations, Debbie Does Due Diligence, Triggered -- to the tenterhooked browser.) Mostly I remember the prof lectured wearing sunglasses, which on more than one occasion concealed a black eye.
Then in senior year there was Herman Melville and the Forms, which was a comparative literature course. Unless that was freshman year, then it was a punk band. I think. TBH, the whole experience is something of a blur. I took classes in economics and FORTRAN, racquetball and contract law, geology and differential equations. I'm not entirely sure what my major was.
Someone could have devoted a lecture to Gershgorin circles, is my point. As I hope I have convinced you, the remainder of my education would not have suffered.
Besides, Gershgorin circles are muy cool.
Here's the deal.
Take any nxn real matrix. You are now going to draw n circles on the real line. Each circle is centered at the one of the diagonal entries of the matrix and the circle radius is equal to the sum of the absolute values of the off-diagonal terms in its row.
Believe it or not, all of the eigenvalues of the matrix live in those circles.
Wait, there's more!
Play the game with the columns instead of the rows. All of the eigenvalues live in that collection of circles, too.
Wait there's more!
Take the intersection of the row circles and the column circles. All of the eigenvalues live in that collection of circles, too.
The game also works if the matrix entries are complex. Now, however, you need to draw your circles in the complex plane rather than on the real line.
All this seems too splendiferous to be true, so I made a Javascript that demonstrates the 2x2 case:
xxx
Enter matrix entries* (or click the "random matrix" button). The app draws two circles and lists the two eigenvalues. Convince your noggin the latter live in the former. (NB: The x-axis is the real axis, but the plane is the complex plane.)
* There's minimal error checking in the code, and by "minimal" I mean none. So don't get sassy with your input.
These are known as Gershgorin circles, after S.A. Gershgorin who described them in 1931. A footnote in Meyer says the idea appeared in previous work by Lévy, Minkowski, and Hadamard, but mathematicians Olga Taussky and Alfred Brauer worked to get Gershgorin's name attached to it because reasons (Lévy, Minkowsky, and Hadamard got other stuff named after them).
In the maelstrom of agony that is mathematics on any given day, such a simple something is a wild gift. A cosmic thing. Useful, too. For example, Gershgorin immediately tells you all diagonally dominant matrices are nonsingular (think about it). Other delights and connections await discovery, should you care to explore.
A proof the circles work as advertised is not terribly complicated, but I'll let you suss it out or look it up yourself.
Gershgorin!
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