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8.5. SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS

Lecture



Having established the existence of a solution to the system of equations

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.1)

it is necessary to determine the nature of the decision: is it unique or is there several solutions, and what kind of solution does it have? The answer to the last question is contained in the following fundamental theorem [4]:

If solving a system of equations   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS exists, then in general it looks like

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.2)

Where   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS - matrix, conventionally inverse with respect to the matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS a   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS - arbitrary size vector   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS .

For proof, multiply both sides of (8.5.2) by the matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS :

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.3)

However, by the condition of the existence of a solution   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . Furthermore, according to the definition of the conditionally inverse matrix,   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . Consequently,   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS and vector   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS is a solution.

Insofar as   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS , then, multiplying both sides of this equality by the matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS get

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.4a)

or

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.4b)

Adding a vector to both parts   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS get

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.5)

This result coincides with the relation (8.5.2) if the vector   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS standing on the right side of formula (8.5.5), replace with an arbitrary vector   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS .

Since the generalized inverse matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS and matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS since the least squares inversions are conditionally inverse, then the general solution of system (8.5.1) can also be represented as

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.6a)

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.6b)

The solution will obviously be the only one   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . Bo all such cases   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . Examining the rank of the matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS , it is possible to prove that [4] if the solution of the system of equations   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS exists that it is unique if and only if the rank of the matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS size   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS equals   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS .

It follows that if a solution to the underdetermined system of equations exists, then it is not unique. On the other hand, an overdetermined system of equations can have only one solution.

Let the exact solution for the system of equations (8.5.1) be obtained. Consider the assessment

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.7)

Where   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS denotes one of the pseudoinverse matrices with respect to   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS which will not necessarily coincide with this solution, since the product of the matrices   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS may not equal the unit matrix. The magnitude of the error, i.e. the deviation of the estimate   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS from true value   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS , usually expressed in terms of the square of the difference of the vectors   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS and   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS in the form of a work

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.8a)

or how

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.8b)

Substituting expression (8.5.7) into (8.5.8a), we obtain

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.9)

Matrix value   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS at which the error (8.5.8) turns out to be minimal, can be found by equating to zero the derivative of the error   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS by vector   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . According to the relation (5.1.34),

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.10)

Equality (8.5.10) is satisfied if the matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS i.e. is a generalized inverse matrix with respect to   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . In this case, the estimation error is reduced to a minimum equal to

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.11a)

or

  8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS (8.5.11b)

As expected, the error becomes zero when   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS . This will happen, for example, if the generalized inverse matrix   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS has rank   8.5.  SOLUTION OF JOINT SYSTEMS OF LINEAR EQUATIONS and is determined by the ratio (8.3.5


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