# Matrix Notation

**1 Matrix Basics
**Recall that we can rewrite the system

using matrix notation as

where

**2 Matrix Operations
**• Addition of matrices.

• Multiplying a matrix by a scalar.

• The product of a matrix and a column vector

• The product of two matrices

• AB = (Ab

_{1},Ab

_{2}).

• In general, AB ≠ BA.

**3 Solving a 2 × 2 Linear System
**Consider the system

with initial conditions x(0) = 1 and y(0) = 1. For

we can guess that the solution is of the form

where λ, c_{1}, c_{2} ∈ R.

**4 Finding Eigenvalues and Eigenvectors
**Let

The system Ax =λx can be written as either

x_{1} + 2x_{2} = λx_{1}

4x_{1} + 3x_{2} = λx_{2}.

We can reduce this system to

(1 −λ)x_{1} + 2x_{2} = 0

(λ^{2} − 4λ − 5)x_{2} = 0.

Therefore, to obtain a nonzero solution either λ = 5 or λ = −1.

• If λ = 5, the first equation in the system becomes −2x_{1}
+x_{2} = 0, and

we can let

• If λ = −1, then

The number λ is called an eigenvalue of A, and x ≠ 0 is an eigenvector

corresponding to λ.

**5 The Principle of Superposition
**The Principle of Superposition tells us that any linear combination of
solutions

to a linear equation is also a solution. Therefore

**6 The Characteristic Equation
**The key to solving the system

is determining the eigenvalues of A. To find this eigenvalues, we need to

derive the characteristic polynomial of A ,

The quantity T = a + d is the sum of the diagonal elements of the matrix

A. We call this quantity the trace of A and write tr(A). Of course, D =

det(A) = ad−bc is the determinant of A. If a 2×2 matrix A has eigenvalues

λ

_{1}and λ

_{2}, then the trace of A is λ

_{1}+ λ

_{2}and det(A) = λ

_{1}λ

_{2}. Indeed, we

can rewrite the characteristic polynomial as

det (A − λI) = λ

^{2}− Tλ + D.

The eigenvalues of A are now given by

We can immediately see that the expression T ^{2} −4D
determines the nature

of the eigenvalues of A.

**7 The Trace-Determinant Plane, D = T ^{2}/4
**

Figure 1: D = T

^{2}/4

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