Mathematics Cheat Sheet For Population Biology (Page 6 of 9) in pdf

Mathematics Cheat Sheet For Population Biology Page 6

Imaginary

(a,b) = a + bi

Real

Figure 2: Argand diagram representing a complex number z = a + bi.

Complex Numbers We encounter complex numbers frequently when we calculate the eigen-

values of projection matrices, so it is useful to know something about them. Imaginary number:

i =

1. Complex number: z = a + bi, where a is the real part and b is a coeﬃcient on the

imaginary part.

It is useful to represent imaginary numbers in their polar form. Deﬁne axes where the

abscissa represents the real part of a complex number and the ordinate represents the imaginary

part (these axes are known as an Argand diagram). This vector, a + bi can be represented by

the angle θ and the radius of the vector rooted at the origin to point (a, b). Using trigonometric

deﬁnitions, a = r sin θ and b = r cos θ, we see that

z = a + ib = r(cos θ + i sin θ).

Believe it or not, this comes in handy when we interpret the transient dynamics of a popu-

lation.

Let z be a complex number with real part a and imaginary part b,

z = a + bi

Then the complex conjugate of z is

¯ z = a

Non-real eigenvalues of demographic projection matrices come in conjugate pairs.

Linear Algebra

A matrix is a rectangular array of numbers

A =

A vector is simply a list of numbers

Mathematics Cheat Sheet For Population Biology Page 6