Basic Linear Algebra Facts for Bio/Math 4230

Matrix Operations

A matrix is a grid of numbers in rows and columns: $${\bf M}=\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right)$$.

When you multiply a matrix by a scalar, that is by a single number or parameter, multiplication is component by component:

$$5 \cdot {\bf M} = 5 \cdot \left( \begin{array}{cc} 1 & 2 \\ ... ...left( \begin{array}{cc} 5 & 10 \\ 15 & 20 \end{array}\right).$$

When you add two matrices you do it component by component:

$$\left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right) +... ...= \left(\begin{array}{cc} 6 & 8 \\ 10 & 12 \end{array}\right).$$

When you multiply a matrix by a vector, however, rows from the left are multiplied by columns on the right:

$$\left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right) \... ...\right) = \left(\begin{array}{c} 17 \\ 39 \end{array}\right) .$$

When matrices are multiplied by matrices, the same rules are followed as with vectors:

$$\left(\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right) \... ...eft(\begin{array}{cc} 19 & 22 \\ 43 & 50 \end{array}\right) .$$

The Identity Matrix

The Identity matrix (I) is the matrix that, when multiplied by either a matrix or vector, doesn't change the result. That is, if M is a matrix and v is a vector, M$\cdot$I = M and I$\cdot$v = v. In two dimensions, $${\bf I} = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$$, and in three dimensions, $${\bf I} = \left(\begin{array}{ccc} 1 & 0 &0\\ 0 & 1& 0 \\ 0 & 0 & 1 \end{array}\right).$$

Equivalence Between Matrices and Linear Equations

Matrix notation is a short-hand way to write down systems of linear equations, as in

$$\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right) ... ...dot x + 2 \cdot y = 5,\\ 3\cdot x + 4 \cdot y = 6. \end{array}$$

Similarly,

$$\left( \begin{array}{ccc} 1 & 2 &3 \\ 4 & 5 & 6 \\ 7 & 8 &... ... = 11, \\ 7 \cdot x + 8 \cdot y + 9 \cdot z = 12 . \end{array}$$

Matrix Multiplication as Mapping in the Plane

Think of $(x,y)$ as the coordinate of a point on a plane. If M $$=\left( \begin{array}{cc} a & b \\ c & d \end{array}\right)$$ is a matrix, then

$${\bf M} \left( \begin{array}{c} x\\ y \end{array}\right) = \... ...}\right) = \left( \begin{array}{c} x'\\ y' \end{array}\right)$$

gives some new coordinates, $(x',y')$, in the plane which are related to the original coordinates. This is what is meant by a map from the original, $(x,y)$, points to the new, $(x',y')$, points. There are only certain kinds of things which can happen with matrix mappings:

• The matrix can stretch or compress the plane in particular directions.
• The matrix can flip or reflect points across a line.
• The matrix can rotate points in the plane.

Eigenvalues and Eigenvectors

The easiest way to think of eigenvalues and eigenvectors is in terms of the stretching and compressing of matrix maps. A matrix, M, can only stretch/compress points in as many directions as there are - in two dimensions there are only two directions, and in three dimensions there are only three possible directions. An eigenvector, v is a direction in which the matrix is a pure stretch, that is M $\cdot$ v = $\lambda$ v, where $\lambda$, the eigenvalue, is the amount of stretching in that direction. Since we do not know, a prioiri what the eigenvalue, eigenvector is, let v = $$\left( \begin{array}{c} x\\ y \end{array}\right)$$. Then

$${\bf M } \cdot {\bf v} = \left( \begin{array}{cc} a & b \\ ... ...( \begin{array}{c} x\\ y \end{array}\right) = \lambda {\bf v}.$$

Since this results (in two dimensions) in two equations with three unknowns, the solution can not be totally determined. Solving for $\lambda$, the eigenvalue, gives an equation

$$\lambda^2 - (a+d) \lambda + (a d - bc) = 0, \quad \mbox{or}\... ... = \frac12 \left[ (a+d) \pm \sqrt{(a+d)^2 - 4 (ad-bc)}\right].$$

Two (parallel) eigenvectors corresponding to this eigenvalue are given by

$${\bf v} = \left( \begin{array}{c} 1\\ \frac{\lambda - a}{b} ... ...d \left( \begin{array}{c} b\\ \lambda - a \end{array}\right) .$$

All eigenvectors for the same eigenvalue are parallel; they can not be solved for any more specifically because the eigenvalue equations are two equations for three unknowns. The eigenvalues relate to matrix mappings:

• Eigenvalues with size bigger than one stretch, while those with size smaller than one compress in the direction corresponding to the eigenvector.
• A negative eigevalue flips or reflects points across a line perpendicular to the corresponding eigenvector.
• If the eigenvalues are complex then the matrix rotates points in the plane.

In higher dimensions eigenvalue/eigenvector pairs satisfy the matrix equation

$${\bf M} {\bf v} = \lambda {\bf v} \quad \mbox{or} \quad ({\bf M} - \lambda{\bf I} ) \cdot {\bf v} = {\bf0}.$$