Inversion of a matrix

Referring back to the set of simultaneous equations at the beginning of this appendix, the objective is usually to solve these for the unknown x terms, lliis is where the use of matrices has a major advantage because referring to equation [E.l] we may rewrite this as 

This equation expresses the solution to the set of simultaneous equations in that each of the unknown x terms is now given by a new matrix [A] multiplied by the known y terms. The new matrix is called the inverse of matrix [A]. The determination of the terms in the inverse matrix is beyond the scope of this brief introduction. Suffice to say that it may be obtained very quickly on a computer and hence the solution to a set of simultaneous equations is determined quickly using equation [E.4], 

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In thi.-. case the adjoint matrix is the same as the matrix of cofactors (as A is a symmetric. njlri.x). The inverse of a matrix is obtained by dividing the elements of the adjoint matrix tlie determinant ... 

Inverse of a Matrix A square matrix A is said to have an inverse if there exists a matrix B such that AB = BA = Z, where Z is the identity matrix of order n. 

If a square matrix has an inverse, the product of the matrix and its inverse equals the unit matrix. The inverse of a matrix A is denoted by A1. 

Thus, if the inverse of a matrix B is known, that of a matrix B — differing from it only by a matrix of rank 1, can be found directly. The matrix of rank 1 could be a single element, or a single row or column. It is possible, for example, to start with the inverse of any matrix (e.g., the identity I), and modify it column by column, each time applying the above formula, until the inverse of A is obtained after at most n steps. In a similar way and with less work, it is possible to build up to A -1A. 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

The inverse of a matrix is the conceptual equivalent to its reciprocal. Therefore if we denote our matrix by X, then the inverse of X is denoted as X-1 and the following relationship holds. 

In Chapter 2, we promised to show the steps involved in taking the inverse of a matrix. Given a 2 x 2 matrix [X]2x2, how is the inverse calculated We can ask the question another way as, What matrix when multiplied by a given matrix [X]rxc will give the identity matrix ([I]) In matrix form we may write a specific example as ... 

The second key point is the accomplishment of a desired goal on the left-hand side of equation 4-8 we have the expression [A] 1 [A], We noted earlier that the key defining characteristic of the inverse of a matrix is that fact that when multiplied by the original matrix (that it is the inverse of), the result is a unit matrix. Thus equation 4-8 is equivalent to... 

But how can we tell then if the answer is correct Well, there is a way, and one that is not too overwhelming. From the definition of the inverse of a matrix, you should obtain a unit matrix if you multiply the inverse of a given matrix by the matrix itself. In our previous chapter [1] we showed this for the 2 x 2 case. For the simultaneous equations at hand, however, the process is only a little more extensive. From the original matrix of coefficients in the simultaneous equations that we are working with, the one called [A] above, we find that the inverse of this matrix is... 

However, having a set of numbers that purports to be the inverse of a matrix, we can verify whether or not it is the inverse of that matrix all we need to do is multiply by the original matrix and see if the result is a unit matrix. We have done this for the 2 x 2 matrix in our previous chapter. An exercise for the reader is to verify that the matrix shown in equation 4-11 is, in fact, the inverse of the matrix [A],... 

To calculate the inverse of a matrix by this procedure is equally tedious and probably more work than solving a set of equations by the brute-force high-school technique. However, the procedure is readily converted into computer code and this is now the only way recommended for matrix inversions. 

It is noted that the inverse of a matrix only exists if >1 0. Any matrix with A = 0 is called singular. 

There is no matrix version of simple division, as with scalar quantities. Rather, the inverse of a matrix (A-1), which exists only for square matrices, is the closest analog to a divisor. An inverse matrix is defined such that AA"1 = A-1 A = I (all three matrices are n X n). In scalar algebra, the equation a-b = c can be solved for b by simply multiplying both sides of the equation by la. For a matrix equation, the analog of solving... 

The inverse of a matrix is defined as a matrix which, when multiplied by the original matrix, gives the identity matrix. This is a diagonal matrix (a square matrix with terms on the diagonal but zeros on all the olT-diagonal positions) with 1 terms on the diagonal. The 2x2 identity matrix is 1 ... 

The inverse of a matrix is usually calculated numerically, particularly in realistically complex engineering applications. Standard computer library subroutines are readily available. We use the IMSL subroutine LEQ2C in this book to calculate the inverse of a complex matrix. 

Remember that the inverse of a matrix has the determinant of the matrix in the denominator of each element. Therefore the denominators of all of the transfer functions in Eq. (15.64) will contain Det [ + Now we know that... 

In Section 5.3 the inverse methods of MLR and PLS/PCR are discussed. The one challenge in using the inverse approach is in the inversion of a matrix. The two approaches discussed for solving the inversion problem are to select variables (MLR) or to estimate factors to use in place of the original measurement variables (PLS/PCR). 

EX114 1.1.4 Inversion of a matrix by Sauss—Jordan elimination see EX112... 

In Examples 1.1.2 and 1.1.3 we did not need the row interchange option of the program. This option is useful in pivoting, a practically indispensable auxiliary step in the Gauss-Jordan procedure, as will be discussed in the next section. Mhile the Gauss-Jordan procedure is a straightforward way of solving matrix equations, it is less efficient than some methods discussed later in this chapter. It is, however, almost as efficient as any other method to calculate the inverse of a matrix, the topics of our next section. 

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