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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 TraceDeterminant Plane, D = T^{2}/4
Figure 1: D = T^{2}/4
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