The Identity Matrix

For a proper choice of boundary conditions, the above mentioned constant matrix can be assumed to be the identity matrix, namely,... 

Here, M is a constant, symmetric positive definite mass matrix. We assume without loss of generality that M is simply the identity matrix I. Otherwise, this is achieved by the familiar transformation... 

Next the error is calculated (Eq. (15), where 1 is the identity matrix). 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

As with the other semi-empirical methods that we have considered so far, the overlap niJtrix is equal to the identity matrix. The following simple matrix equation must then be solved ... 

The set of eigenveetors of any Hermitian matrix form a eomplete set over the spaee they span in the sense that the sum of the projeetion matriees eonstrueted from these eigenveetors gives an exaet representation of the identity matrix. 

It turns out that not only the identity matrix I but also the matrix M itself ean be expressed in terms of the eigenvalues and eigenveetors. In the so-ealled speetral representation of M, we have... 

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 det C 0, C exists and can be found by matrix inversion (a modification of the Gauss-Jordan method), by writing C and 1 (the identity matrix) and then performing the same operations on each to transform C into I and, therefore, I into C". ... 

C CT] is known as the pseudo inverse of C. Since the product of a matrix and its inverse is the identity matrix, [C CT][C CT] disappears from the right-hand side of equation [32] leaving... 

A unit matrix is a diagonal matrix in which all of the diagonal elements are equal to 1. The unit matrix is sometimes callesd the identity matrix. It is often denoted as I. 

Here we must regard and as forming one-column matrices. Note that the square matrices (.gk h,y and are mutually reciprocal, and that their product is the identity matrix ... 

Unconstrained ML for Gaussian white noise. For Gaussian stationary noise, the covariance matrix is diagonal and proportional to the identity matrix ... 

In the case in which the errors are independent of each other their covariances will be zero, and if they also have the same variance, then D = oH, with the constant being the common variance and I being the identity matrix. In this case, the same 0 minimizing (Eq. 3.3) would also minimize (Eq. 3.2) and the OLSE can therefore be seen as a particular case of the WLSE. 

The barred matrices have the same properties as those of eqn 5 in the case of C normalization to unity of the single columns is ensured by an ad hoc diagonal matrix N. As will appear below (eqn 6), if terms in AC of order higher than the first are negligible, N can be taken equal to the identity matrix. This is what will be assumed... 

Time-invariant systems can also be solved by the equations given in Table 41.10. In that case, F in eq. (41.15) is substituted by the identity matrix. The system state, x(j), of time-invariant systems converges to a constant value after a few cycles of the filter, as was observed in the calibration example. The system state. 

As seen by comparing Equations 5.6 and 5.12 the steepest-descent method arises from Newton s method if we assume that the Hessian matrix of S(k) is approximated by the identity matrix. 

To initialize the algorithm we start with our best initial guess of the parameters. 0O. Our initial estimate of the covariance matrix P0 is often set proportional to the identity matrix (i.e., P0 =y2l). If very little information is available about the parameter values, a large value for y2 should be chosen. 

The values of the elements of the weighting matrices R, depend on the type of estimation method being used. When the residuals in the above equations can be assumed to be independent, normally distributed with zero mean and the same constant variance, Least Squares (LS) estimation should be performed. In this case, the weighting matrices in Equation 14.35 are replaced by the identity matrix I. Maximum likelihood (ML) estimation should be applied when the EoS is capable of calculating the correct phase behavior of the system within the experimental error. Its application requires the knowledge of the measurement... 

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