Lesson 24, Matrices and Linear Systems

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Handout Lesson 24, Matrices and Linear Systems

Textbook Section(s).

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This lesson is based on Section 5.1 of your textbook by Edwards, Penney, and Calvis.

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Matrices.

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Before coming to class today, you were expected to work through the Supplemental Notes on Matrices material found at the top of the module in Brightspace-> Content-> Lecture Materials. These videos and notes contained information about:

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component wise operations on matrices
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matrix representations for systems of linear equations and row operations for solving systems of linear equations
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matrix multiplication
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inverses of square matrices, including an algorithm for calculating the determinant of a \(2 \times 2\) matrix
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determinants
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Any questions?

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Matrix notation for first-order linear systems of differential equations.

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A first-order linear system of differential equations can be represented as

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\begin{gather*} \frac{d\mathbf{x}}{dt}=\mathbf{P}(t)\mathbf{x}+\mathbf{f}(t) \end{gather*}

where \(\mathbf{P}(t)\) is a matrix and \(\frac{d\mathbf{x}}{dt}\text{,}\) \(\mathbf{x}\text{,}\) and \(\mathbf{f}(t)\text{,}\) all of compatible sizes. NOTE: When writing, it is difficult to use boldface for matrices and vectors. It is standard practice to use capital letters for matrices. For vectors, I will sometimes use the arrow notation \(\overrightarrow{x}\) when writing to emphasize that I am talking about a vector.

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Example 154. Using matrix notation for systems of DE’s.

Rewrite the first-order linear system of differential equations using matrix notation. Then find the associated homogeneous equation for the system.

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\begin{align} \begin{cases} x_1' \amp = e^tx_1+\cos(t)x_2+t^2+1 \\ x_2' \amp = \sin(t)x_1+7x_2+5t \\ \end{cases}\tag{#} \end{align}

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Principle of Superposition.

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Theorem 155.

Let \(\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}\) be solutions of the homogeneous system of linear differential equations

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\begin{gather} \frac{d\mathbf{x}}{dt}=\mathbf{P}(t)\mathbf{x}\tag{✶} \end{gather}

on the open interval \(I\text{.}\) Then for constants \(c_1, c_2, \dots, c_n\text{,}\)

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\begin{gather*} \mathbf{x}(t)=c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t)+ \dots + c_n\mathbf{x_n}(t) \end{gather*}

is also a solutions to (✶).

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Linear independence of vector functions and the Wronskian.

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Definition 156.

\(\mathbf{x_1}(t), \mathbf{x_2}(t), \dots, \mathbf{x_n}(t)\) are linearly dependent if there are constants \(c_1, c_2, \dots, c_n\) that are not all zero such that

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\begin{gather} c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t)+ \dots + c_n\mathbf{x_n}(t) = \mathbf{0}\tag{†} \end{gather}

If the only set of constants that satisfy (†) is \(c_1 = c_2 = \dots = c_n=0\text{,}\) then the vector valued functions \(\mathbf{x_1}(t), \mathbf{x_2}(t), \dots, \mathbf{x_n}(t)\) are linearly independent

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.

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Definition 157.

Let \(\mathbf{x_1}(t), \mathbf{x_2}(t), \dots, \mathbf{x_n}(t)\) be \(n\) vector valued functions, each with \(n\) components, so that

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\begin{gather*} \mathbf{x_j}(t)=\left[ \begin{array}{c} x_{1,j}(t) \\ x_{2,j}(t) \\ \vdots \\ x_{n,j}(t) \end{array} \right] \end{gather*}

(NOTE: In this notation, the second index indicates the number of the vector valued function.)

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Then the Wronskian of \(\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}\) is

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\begin{align*} W(t) \amp = W(\mathbf{x_1}, \mathbf{x_2}, \dots \mathbf{x_n})\\ \amp = \det\bigg(\bigg[ \mathbf{x_1}(t) \quad \mathbf{x_2}(t) \quad \dots \quad \mathbf{x_n}(t) \bigg] \bigg)\\ \amp = \left| \begin{array}{cccc} x_{1,1}(t) \amp x_{1,2}(t) \amp \dots \amp x_{1,n}(t) \\ x_{2,1}(t) \amp x_{2,2}(t) \amp \dots \amp x_{2,n}(t) \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ x_{n,1}(t) \amp x_{n,2}(t) \amp \dots \amp x_{n,n}(t) \end{array} \right| \end{align*}

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Theorem 158. The Wronskian and linear independence.

Let \(\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}\) be \(n\) solutions of the homogeneous system of linear differential equations

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\begin{gather} \frac{d\mathbf{x}}{dt}=\mathbf{P}(t)\mathbf{x}\tag{#} \end{gather}

where \(\mathbf{P}(t)\) is an \(n \times n\) matrix. Suppose \(\mathbf{P}(t)\) is continuous on the open interval \(I\text{.}\)

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If \(\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}\) are linearly dependent on \(I\) then
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\begin{gather*} W(\mathbf{x_1}, \mathbf{x_2}, \dots \mathbf{x_n})(t)=0 \end{gather*}

for all \(t \in I\text{.}\)
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If \(\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}\) are linearly independent on \(I\) then
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\begin{gather*} W(\mathbf{x_1}, \mathbf{x_2}, \dots \mathbf{x_n})(t)\neq 0 \end{gather*}

for all \(t \in I\text{.}\)
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Example 159. The Wronskian, general solutions, and and IVP.

(Based on number 25 from Section 5.1 of your textbook by Edwards, et.al.)

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Let

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\begin{gather*} \mathbf{x_1}(t)=\left[ \begin{array}{c} 3e^{2t} \\ 2e^{2t} \end{array} \right] \qquad \text{ and } \qquad \mathbf{x_2}(t)=\left[ \begin{array}{c} e^{-5t} \\ 3e^{-5t} \end{array} \right] \end{gather*}

and consider the system

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\begin{align} \mathbf{x}'=\left[ \begin{array}{cc} 4 \amp -3 \\ 6 \amp -7 \end{array} \right] \mathbf{x}\tag{✶✶} \end{align}

Show that \(\mathbf{x_1}\) and \(\mathbf{x_2}\) are solutions to (✶✶).
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Use the Wronskian to show that \(\mathbf{x_1}\) and \(\mathbf{x_2}\) are linearly independent.
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Find a general solution to (✶✶).
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Find the particular solution of (✶✶) that satisfies the initial condition \(\mathbf{x}(0)=\left[\begin{array}{c} 8 \\ 0 \end{array}\right]\)
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Solutions of nonhomogeneous linear systems of differential equations.

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Theorem 160.

Let \(\mathbf{P}(t)\) be an \(n \times n\) matrix and let \(\mathbf{x_p}(t)\) be a particular solution of

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\begin{gather} \frac{d\mathbf{x}}{dt}=\mathbf{P}(t)\mathbf{x}+\mathbf{f}(t)\tag{#} \end{gather}

on and open interval \(I\) on which \(\mathbf{P}(t)\) and \(\mathbf{f}(t)\) are continuous. Let \(\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}\) be linearly independent solutions of the associated homogeneous system of linear differential equations on \(I\text{.}\) If \(\mathbf{x}(t)\) is any solution of (#), then there are numbers \(c_1, c_2, \dots, c_n\) such that

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\begin{gather*} \mathbf{x}(t)=\mathbf{x_p}(t)+c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t)+ \dots + c_n\mathbf{x_n}(t) \end{gather*}

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