Projection Matrices

By extending this to include vectors for all three translation and rotational modes, a projection matrix is obtained. 

The geometric meaning of Eq. (35.14) is that the best fit is obtained by projecting all responses orthogonally onto the space defined by the columns of X, using the orthogonal projection matrix X(X X) X (see Section 29.8). 

This number is the answer to the question originally posed. This is the number of real conditions required to fix experimentally a complex, normalized, hermitian, projection matrix. For example, this number of experimental structure factors, Equation (1), would suffice to fix P Equation (6). 

An elegant classification strategy using projection matrices was proposed by Crowe et al. (1983) for linear systems and extended later (Crowe, 1986, 1989) to bilinear ones. Crowe suggested a useful method for decoupling the measured variables from the constraint equations, using a projection matrix to eliminate the unmeasured process variables. 

For linear plant models Crowe et al. (1983) used a projection matrix to obtain a reduced system of equations that allows the classification of measured variables. They identified the unmeasured variables by column reduction of the submatrix corresponding to these variables. 

A projection matrix P was defined by Crowe, such that premultiplying the Jacobian matrix A2 with P yields... 

In order to obtain the projection matrix P, Crowe proposed the following procedure ... 

In this chapter, the use of projection matrix techniques, more precisely the Q-R factorization, to analyze, decompose, and solve the linear and bilinear data reconciliation problem was discussed. This type of transformation is selected because it provides a very good balance of numerical accuracy, flexibility, and computational cost (Goodall, 1993). 

Now consider an nx(n m) projection matrix Z that has the property Z Vh = 0. This matrix can be created either by an orthonormal factorization of Vh or simply by partitioning the x variables into u and v (decision and dependent variables, respectively) and into Z is then given... 

This construction in which a vector is used to form a matrix v(i)Xv(i) is called an "outer product". The projection matrix thus formed can be shown to be idempotent, which means that the result of applying it twice (or more times) is identical to the result of applying it once P P = P. This property is straightforward to demonstrate. Let us consider... 

We define a normalized column vector g and a projection matrix P by... 

The derivation of the eigenphase sum for the Simonius S matrix, SSim(E), is straightforward by generalizing the procedure for an isolated resonance shown in Section 2.2.2. Compared with the Breit-Wigner S matrix, Sm(E), the matrix Sr in Eq. (34) is now replaced by SPN, or the product of matrices Sv/ each having the same apparent form as Sr but with different resonance parameters and a different projection matrix. Since the determinant of the... 

There seems no simple unitary transform of the time-delay matrix QSim, such as Eq. (72), if more than two resonances overlap each other. Instead, we may take advantage of the rank 1 of the projection matrix Pv of Eq. (61) in Qv of Eq. (69). Thus, for the eigenvalue equation... 

Before selecting a certain field screening kit, test it on the project matrix and compare results to laboratory data for the same samples. 

The leverage, / , of the z th calibration sample is the z th diagonal of the hat matrix, H. The leverage is a measure of how far the z th calibration sample lies from the other n - 1 calibration samples in X-space. The matrix H is called the hat matrix because it is a projection matrix that projects the vector y into the space spanned by the X matrix, thus producing y-hat. Notice the similarity between leverage and the Mahalanobis distance described in Chapter 4. 

We now define the form of the projection matrix P. Extending equation (4.73) to include perturbations arising from physical processes we get... 

The matrix formed from the product of vectors, P = u (u ), is called a vector outer product. The expansion of a matrix in terms of these outer products is called the spectral resolution of the matrix. The matrix P satisfies the relation pkpt pk (Jq matrices of the more general form, P = X] P , where the summation is over an arbitrary subset of outer product matrices constructed from orthonormal vectors. Matrices that satisfy the relation P = P are called projection operators or projection matrices If P is a projection matrix, then (1 - P) is also a projection matrix. Projection matrices operate on arbitrary vectors, measure the components within a subspace (e.g. spanned by the vectors u used to define the projection matrix) and result in a vector within this subspace. 

Finally the state Hessian matrix M is seen from Eq. (149) to be proportional to the representation of the Hamiltonian operator in the orthogonal complement basis, but with all the eigenvalues shifted by the constant amount (0). The dimension of the matrix M will be one less than the length of the CSF expansion unless it is constructed in the linearly dependent projected basis or the overcomplete CSF expansion set basis. Since the Hamiltonian matrix must usually be constructed in the CSF basis in the MCSCF method anyway, it is most convenient if M and C are also constructed in this basis. The transformation to the projected basis, if explicitly required, involves the projection matrix (1 — cc ). The matrix M only requires the two-electron integral subset that consists of all four orbital indices corresponding to occupied orbitals. 

Projection matrix Sample mean of variable x Process variables data matrix (x x m) Quality variables data matrix (x x q)... 

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