7.2 Review of Matrices

The results of matrix theory2 should be brought to bear on the initial value problem for a system of linear differential equations for both theoretical and computational reasons.

A brief summary of the facts that will be needed later is what this section and the next are devoted to.
There are more details in any elementary book.
We assume that you know how to evaluate determinants.

In this section, we assume that the elements of matrices may be complex numbers, because we assume that the elements of certain matrices are real numbers.

The English mathematician Arthur Cayley published a paper on the properties of matrices in 1858, but the word matrix was introduced by his friend James Sylvester in 1850.

After practicing law from 1849 to 1863, Cayley did some of his best mathematical work and became a professor of mathematics at Cambridge, where he held the position for the rest of his life.
The development of matrix theory was rapid after Cayley's work.

matrix multiplication is not commutative.

The definition of multiplication is given in Eq.

There is a product that is defined for any two vectors with the same number of components.

The product is a complex number and can be compared.

It may not be a real number.

If one of the matrices is the identity, the commutative law holds.

The use of determinants is one way.

It's by means of elementary row operations.

The inverse matrix calculation is shown in this way.

In calculating inverses, row reduction methods are preferred.

Below the statement is the result of each step.

The definitions are expressed.

In this section, we look at some results from linear algebra that are important for the solution of linear differential equations.

The solution can be found by subtracting each side of Eq.

It's not true that -1 does not exist.

There is a more complicated situation for the nonhomogeneous system.

The system has many solutions if condition 5 is met.

The resemblance between Eq.

and another is noted.

The results in the preceding paragraph are important for figuring out the solutions of linear systems.

It is best to use row reduction to transform the system into a simpler one from which the solution can be written down easily.

The equals sign is said to be replaced by a dashed line.

It is easy to see if the system has solutions after this is done.

Elementary row operations on the augmented matrix correspond to legitimate operations on the equations in the system.
The process is illustrated by the following examples.

We can see zeros in the lower left part of the matrix with row operations.

The result is recorded below.

Since the solution is unique, we conclude that the coefficients are nonsingular.

It is possible to show that the condition is just Eq.

We can solve the system by choosing one of the unknowns and then solving it for the other two.

The second term is on the right side of the book.

Row reduction can be used to solve systems in which the number of equations is different from the number of unknowns.

There are nonzero solutions.

Find a linear relation among them if they are linearly dependent.

The problem of finding the axes of stress in an elastic body and the modes of free vibration in a conservative system is encountered.

This example shows how eigenvalues and eigenvectors are found.

To find the eigenvectors, we have to return to Eq.

This fact may lead to problems later on in the solution of systems of differential equations.

The double eigenvalue is associated with two linearly independent eigenvectors.

All eigenvalues are true.

If all eigenvalues are simple, the associated eigenvectors form a set of vectors.

The discussion in this section is the same as in Sections 3.2, 3.3, and To discuss the system most effectively, we write it in matrix notation.

The similarity between systems of equations and single (scalar) equations is emphasized by the use of vectors and matrices.

This is enough to guarantee the existence of solutions.

The nonhomogeneous equation is one of the methods that can be used to solve the homogeneous equation.

The structure of solutions of the system is stated in the main facts.

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By applying Theorem 7.4.1, it follows that every finite linear combination of solutions of Eq.

The question is whether all solutions of Eq.

All of the expressions of the form are solutions of the system, according to Theorem 7.4.1, and all of the solutions of Eq.

are also solutions of the system.

The system (3) always has at least one fundamental set of solutions according to the next theorem.

Once a set of solutions has been found, other sets can be created by forming linear combinations of the first set.

The set given by Theorem 7.4.4 is usually the simplest.

Every solution of the system can be represented as a linear combination of any fundamental set of solutions under the conditions given in this section.

There are two problems indicating an alternative derivation of 7.42.

It is an equilibrium solution.

Other solutions leave from it.
The situation is more complicated for higher order systems.

A direction field can be used to gain a qualitative understanding of solutions.

There are more precise results from including some solution curves in the plot.
A plot that shows a sample of the trajectory of a system is called a Examples of direction fields and phase portraits occur later in the section.

The general solution of the system is constructed by analogy with the treatment of second order linear equations in Section 3.1.

We seek solutions.

Substituting from Eq.

To solve the system of differential equations, we need to solve the system of algebraic equations.

Two examples show the solution procedure for 2 x 2 coefficients matrices.

We show how to make the portraits.

Determine the qualitative behavior of solutions by Plotting a direction field.

Find the general solution and then draw a series of arcs.

A typical solution leaves the neighborhood of the origin and has a slope of 2 in either the first or third quadrant.

If the determinant of coefficients is zero, the equations have a nontrivial solution.

In this case, the origin is called.

We have described how to draw a qualitatively correct sketch of the trajectory of a system in the preceding paragraph.

To produce a detailed and accurate drawing, such as and other figures that appear later in this chapter, a computer is extremely helpful.

Draw a direction field for this system, then find its general solution and plot several trajectories in the phase plane.

We have a combination of the two fundamental solutions.

The origin is a part of the system.

The trajectory would be the same if the eigenvalues were positive rather than negative.

If the eigenvalues are positive, the nodes are stable.

Two main cases for 2 x 2 systems have eigenvalues that are real and different, either the same sign or opposite sign.

The assumption was made at the beginning of the section.

Returning to the general system, we follow the examples.

The general solution of the system is determined by the nature of the eigenvalues and corresponding eigenvectors.

All eigenvalues are not the same.

There are eigenvalues in complex conjugate pairs.

Some values are repeated.

The exponential function is never zero.

The solutions of the differential system are given by Eq.

This case is shown in the following example.

The coefficients matrix is real and symmetric.

The initial conditions are critical to the solution's behavior.

The solutions arise from complex eigenvalues.

Section 3.4 states that it is possible to get a full set of real-valued solutions.
Section 7.6 talks about this.

If an eigenvalue is repeated, there can be more serious difficulties.

The number of linearly independent eigenvectors may be smaller than the number of eigenvalue.
It is necessary to find additional solutions of another form to construct a fundamental set of solutions.
Section 7.8 deals with the case of repeated eigenvalues.

If the eigenvalues are different, the solutions of the differential equation are complex-valued.

Plot a few trajectory of the system and draw a direction field.

Consider the previous system of differential equations.

Section 3.4 states that we can find two real-valued solutions.

To simplify the calculations and to allow easy visualization of the solutions in the phase plane.

Show them graphically.

The trajectory of the plane is suggested in the plot.

For a system with a positive real part, the trajectory is similar to those in but the direction of motion is away from the origin.

The origin is not stable.

The origin is said to be stable, but not asymptotically stable.

The current is 2 amperes and the voltage is 2 volts.

The real and imaginary parts of the solution form a pair of linearly independent real-valued solutions.

The coefficients of the matrix control the behavior of the trajectory.

The eigenvalues can change from real to complex.

Both eigenvalues are negative, so all trajectory approach the origin.
The eigenvalues are complex and the trajectory are spirals.
The case is different and the origin is not stable.
There is a critical value where the direction of the spirals changes.
Eigenvalues are real and equal.
The origin is the same as before, but the phase portrait is different.
Section 7.8 is where we take this case.

The three main cases that can occur are for second order systems with real coefficients.

Transitions between two of the cases just listed make other possibilities less important.

A zero eigenvalue occurs during the transition between a saddle point and a nodes.
There is a transition between stable and unstable spiral points.
The real and equal eigenvalues appear during the transition between the spiral points.

The idea of a fundamental matrix can illuminate the structure of the solutions of linear differential equations.

The columns of a fundamental matrix are linearly independent.

A fundamental matrix can be used to write the solution of an initial value problem.

To solve a given initial value problem one would normally solve Eq.

The two sides of Eq are related.

A given physical system can be started from many different initial states.

By comparing the numbers.

We are now looking at this possibility.

Some or all of the equations involve more than one of the unknown variables.

Each equation can be solved independently of all the others, which is a much easier task, if each equation involves only a single variable.

Such a transformation can be accomplished with ennivectors.

It's very simple.

The same result was obtained in Section 7.5.

It is necessary to calculate the eigenvalues and eigenvectors of the coefficients in the system of differential equations if this diagonalization procedure is used.
The problem of diagonalizing a matrix is the same as the problem of solving a system of differential equations.

Obtain a fundamental matrix for the system and transform it to a fundamental matrix for the original system.

It follows from the beginning.

The method of successive approximations can be applied to systems of equations.

This possibility is shown in the following example.

We need to return to Eq.

to determine the eigenvectors.

There is only one linearly independent eigenvector associated with the double eigenvalue.

To construct the general solution.

An example is the first thing we consider.

All nonzero solutions leave from the origin.

There is no nonzero solution of the system.

The system is solvable because the second row is proportional to the first.

The graph of the solution is more difficult to analyze than other examples.

In this case, the origin is an improper one.

The trajectory are similar if the eigenvalues are negative.
If the eigenvalues are positive or negative, an improper node is stable.

There is a difference between a system of two first order equations and a single second order equation.

When there is a double eigenvalue and a single associated eigenvector, example 2 is typical of the general case.

One solution is similar to Eq.

It can be shown that it is possible to solve Eq.

Section 7.7 explains that fundamental matrices are formed by arranging linearly independent solutions in columns.

If we start from the beginning again.

DropCatch Jordan, a professor at the E'cole Polytechnique and the College de France, made important contributions to analysis.

The equations can be solved in reverse order.

The general solution of Eq.

is the same argument as in Section 3.6.

The solutions of the nonhomogeneous system are the remaining terms.

Coefficients are not determined.

The method of undetermined coefficients is a second way to find a solution to the non homogeneous system.

To use this method, one must assume the form of the solution with some coefficients and then try to find the coefficients that will satisfy the differential equation.
The correct form of the solution can be predicted in a simple and systematic manner.
The procedure for choosing the form of the solution is the same as for linear second order equations.

By changing Eq.

We find out that the second one.

The solution is not the same as the one contained in the book.

There are more general problems in which the coefficients are not constant or diagonalizable.

The general solution of the nonhomogeneous system is constructed using the method of variation of parameters.

Every solution of the system is contained in the expression.

If we choose for the particular solution in Eq, we can write the solution of this problem more quickly.

The general solution of the system was given.

There is a problem Solving Eq.

The methods for solving nonhomogeneous equations have advantages and disadvantages.

The solution of several sets of equations may be involved in the method of undetermined coefficients.

The method of diagonalization requires the inverse of the transformation matrix and the solution of a set of uncoupled first order linear equations.

The inverse of the transformation matrix can be written down without calculation, a feature that is more important for large systems.
The most general method is variation of parameters.

It involves the solution of a set of linear equations with variable coefficients, followed by an integration and a matrix multiplication, so it may be the most complicated from a computational viewpoint.

There may be little reason to choose one of the methods over the other for a small system with constant coefficients.
The method of diagonalization is more complicated if the coefficients are not diagonalizable, but only reducible to a Jordan form, and the method of undetermined coefficients is only practical for nonhomogeneous terms.

The Laplace transform can be used for initial value problems for linear systems with constant coefficients.

We don't give any details since it's used the same way as in Chapter 6.

In any introductory book on the subject, there is further information on matrices and linear algebra.