Square matrix inverse

To compute the inverse of a square matrix it is necessary to first calculate its determinant, A. The determinants of 2 x 2 and 3x3 matrices are calculated as follows ... 

The degree of the least equation, k, is called the rank of the matrix A. The degree k is never greater than n for the least equation (although there are other equations satisfied by A for which k > n). If A = n, the size of a square matrix, the inverse A exists. If the matrix is not square or k < n, then A has no inverse. 

The matrix (C) is called the generalized inverse of C. Having estimated the matrix K, one can then estimate the amounts of analytes in an unknown sample. If the number of sensors is equal to the number of analytes, iCis a square matrix. If K exists then... 

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. 

The inverse of a square matrix is denoted by a superscript -1 and is defined as... 

Matrix division is not defined, although if C is a square matrix, C (the inverse of C) can usually be defined so that... 

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. 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

If we consider the relative merits of the two forms of the optimal reconstructor, Eq. s 16 and 17, we note that both require a matrix inversion. Computationally, the size of the matrix inversion is important. Eq. 16 inverts an M x M (measurements) matrix and Eq. 17 a P x P (parameters) matrix. In a traditional least squares system there are fewer parameters estimated than there are measurements, ie M > P, indicating Eq. 16 should be used. In a Bayesian framework we are hying to reconstruct more modes than we have measurements, ie P > M, so Eq. 17 is more convenient. 

Any non-singular square matrix A possesses an inverse matrix A defined as... 

The above equations are developed using the theory of least squares and making use of the matrix inversion lemma... 

Equations 13.14 to 13.16 constitute the well known recursive least squares (RLS) algorithm. It is the simplest and most widely used recursive estimation method. It should be noted that it is computationally very efficient as it does not require a matrix inversion at each sampling interval. Several researchers have introduced a variable forgetting factor to allow a more precise estimation of 0 when the process is not "sensed" to change. 

The inverse, A 1, of a matrix A is defined by the celadon AA = E. If A is a square matrix, its inverse may exist - although not necessarily so. This question is addressed later in this section. Rectangular, nonsquace, matrices may... 

In principle, the relationships described by equations 66-9 (a-c) could be used directly to construct a function that relates test results to sample concentrations. In practice, there are some important considerations that must be taken into account. The major consideration is the possibility of correlation between the various powers of X. We find, for example, that the correlation coefficient of the integers from 1 to 10 with their squares is 0.974 - a rather high value. Arden describes this mathematically and shows how the determinant of the matrix formed by equations 66-9 (a-c) becomes smaller and smaller as the number of terms included in equation 66-4 increases, due to correlation between the various powers of X. Arden is concerned with computational issues, and his concern is that the determinant will become so small that operations such as matrix inversion will be come impossible to perform because of truncation error in the computer used. Our concerns are not so severe as we shall see, we are not likely to run into such drastic problems. 

Now that we have a correct equation, we want to solve this equation (or equation 69-3, which is essentially equivalent) for b. Now, if matrix A had the same number of rows and columns (a square matrix), we could form its inverse, and multiply both sides of equation 69-3 by A-1 ... 

The matrix formed by multiplying A by its transpose AT is a square matrix, and therefore may be inverted. Therefore, if we multiply both sides of equation 69-8 by the matrix inverse of ATA, we have... 

Where, in this whole derivation, did the question of least squares even come up, much less show that equation 69-10 represents the least-squares solution All we did was a formalistic manipulation of a matrix equation, in order to allow us to create some necessary intermediate matrices, and in a form that would permit further computations, specifically, a matrix inversion. 

An equivalent statement is that no row of the coefficient matrix (j8) can be formed as a linear combination of the other rows. Since the matrix s determinant is nonzero when and only when this statement is true, we need only evaluate the determinant of (/ ) to demonstrate that a new basis B is valid. In practice, this test can be accomplished using a linear algebra package, or implicitly by testing for error conditions produced while inverting the matrix, since a square matrix has an inverse if and only if its determinant is not zero. 

Note that the denominator of (A. 17), the determinant of A = A, is a scalar. If A = 0, the inverse does not exist. A square matrix with determinant equal to zero is called a singular matrix. Conversely, for a nonsingular matrix A, det A 0. 

To solve the least squares problem for the estimate of the measurement errors we need to invert the covariance matrix <. It is possible to relate to through a simple recursive formula. Let us recall the following matrix inversion lemma (Noble, 1969) ... 

The left and right product of a square matrix with its inverse results in an identity matrix. In Matlab the command inv(X) or equivalently XA(-1) is used for matrix inversion. Only square matrices can be inverted. ... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

It is important to stress that for this to work, the independently known matrix A of absorptivity coefficients needs to be square, i.e. it has previously been determined at as many wavelengths as there are chemical species. Often complete spectra are available with information at many more wavelengths. It would, of course, not be reasonable to simply ignore this additional information. However, if the number of wavelengths exceeds the number of chemical species, the corresponding system of equations will be over determined, i.e. there are more equations than unknowns. Consequently, A will no longer be a square matrix and equation (2.22) does not apply since the inverse is only defined for square matrices. In Chapter 4.2, we introduce a technique called linear regression that copes exactly with these cases in order to find the best possible solution. 

The pseudo-inverse for the calculation of the shift vector in equation (4.67) has been computed traditionally as J+= (J Jp1 J. Adding a certain number, the Marquardt parameter mp, to the diagonal elements of the square matrix J J prior to its inversion, has two consequences (a) it shortens the shift vector 8p and (b) it turns its direction towards steepest descent. The larger the Marquardt parameter, the larger is the effect. In matrix formulation, we can write ... 

Matrix inversion is analogous to division. Multiplication of A with its inverse A-1 gives an identity matrix, / (see Figure A.2.6). The inverse is only defined for square matrices that are not singular. A matrix is singular if at least one row (or column) contains equal numbers, or at least one column (or row) is a linear combination of... 

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 ... 

If the determinant of the matrix to be inverted is zero, the calculations to be performed are undefined. This suggests a general rule a square matrix has an inverse if and only if its determinant is not equal to zero. A matrix having a zero determinant is said to be singular and has no inverse. As an example of matrix inversion, consider the 2x2 matrix... 

One sees that the ion flow caused by a gas is proportional to the partial pressure. The linear equation system can be solved only for the special instance where m = g (square matrix) it is over-identified for m> g. Due to unavoidable measurement error (noise, etc.) there is no set of overall ion flow Ig (partial pressures or concentrations) which satisfies the equation system exactly. Among all the conceivable solutions it is now necessary to identify set 1 which after inverse calculation to the partial ion flows 1, will exhibit the smallest squared deviation from the partial ion currents i actually measured. Thus ... 

Consider the n h square matrix A and find its inverse A defined by... 

Example 1.1.4 Inversion of a square matrix try Gauss-Jordan elimination. 

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