Matrices identity

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. 

We shall often encotmter square matrices, which have the same number of rows and columns. A diagonal matrix is a square matrix in which all the elements are zero except for those on the diagonal. The unit or identity matrix I is a special type of diagonal matrix in which all the non-zero elements are 1 thus the 3x3 unit matrix is ... 

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... 

Equation 11.16 can be solved for the metal ion s concentration if its activity coefficient is known. This presents a serious complication since the activity coefficient may be difficult to determine. If, however, the standards and samples have an identical matrix, then yM + remains constant, and equation 11.16 simplifies to... 

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. 

I = identity matrix, which has ones on diagonal, zeros elsewhere M = mih matrix, which transforms mill-feed-size distrihiition into mill-product-size distrihiition... 

Before equations (9.99) can be run, and initial value of P(/c//c) is required. Ideally, they should not be close to the final value, so that convergence can be seen to have taken place. In this instance, P(/c//c) was set to an identity matrix. Figure 9.16 shows the diagonal elements of the Kalman gain matrix during the first 20 steps of the recursive equation (9.99). 

Identity matrix This is a diagonai matrix with aii diagonai eiements equai to unity, and is normaiiy denoted by I. 

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... 

The following 3x3 matrices illustrate a square, a diagonal and an identity matrix ... 

Multiplication of an nxp matrix X with an identity matrix leaves the original matrix unchanged ... 

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