(no subject)

Let ∑ be a symmetric, positive definite matrix. I have in mind a covariance matrix.

Diagonalize it:

∑ = P D P⁻¹

Since it’s symmetric, P⁻¹ = P’ (by which I mean, P transpose):

∑ = P D P’

Since it’s positive definite, all the elements in the diagonal matrix D are positive, so you can express D as the square of another diagonal matrix:

D = Q²

∑ = P Q² P’

Define

A = P Q P’

Note that A is symmetric.

Then

A’ A = P Q P’ P Q P’ = P Q² P’ = ∑

This means that quadratic forms of ∑ can be expressed as the lengths of vectors after being transformed by A:

|| A v ||² = v’ A’ A v = v’ ∑ v

I was thinking about this because of the optimization problem

Maximize u’ ∑ u subject to ||u|| = 1

In two dimensions, I can visualize this. The set allowed by the constraint is a circle. The image of that circle under A is an ellipse, since I interpret P and P’ as rotation matrices, and Q as scaling the x and y axes. Take a vector pointing to a point on the circle, rotate it clockwise, scale the axes, rotate it back counterclockwise.

Since u’ ∑ u = || A u ||², our goal is to get the point on the ellipse that is as far as possible from the center. So we want a vector Au that points along the major axis of the ellipse. And our maximum possible value will be the squared length of this vector.

It turns out this comes from a vector u which points along the major axis--this vector is an eigenvector, and it points along the major axis before and after the transformation. I came to that conclusion thinking about what A = P Q P’ does in three parts: rotate clockwise, scale the axes, rotate counterclockwise.

(no subject)

Watson said we should be aiming for mechanisms with a few genes that explain like half of what goes on.

Simple models have the virtue of being able to deduce an answer form them. It may or may not be right, but you can check. If it is right then maybe you are surprised.

A complex model, if you can’t figure out what it implies, or if everything is possible, can you ever surprise yourself by thinking about it?

(no subject)

new cards don’t have the regenerate keyword?

I always took it for granted but it was actually kind of confusing. I fit would die later in the turn, it doens’t, and instead remove it from combat, remove all damage, and tap it?

Apparently now htey just have ‘indestructible until end of turn.’ That’s better.

When I first learned regenerate was gone it was like the floor had been taken out from under me. The Apprentice guy is president. The world I grew up in is gone. But, at least as far as Magic: The Gathering keyword effects goes, the new world makes more sense now.

(no subject)

tortoise like, “that human is actually like 150 in tortoise years”

Cat

Cat comes into play with 8 life tokens.

Remove a life token from Cat: Regenerate Cat.