Matrix of cofactors

The transpose of a square matrix is, of course, another square matrix. The transpose of a symmetric matrix is itself. One particularly important transpose matrix is the adjoint natris, adJA, which is the transpose matrix of cofactors. For example, the matrix of cofactors ul liie 3x3 matrix... 

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 ... 

The cofactor matrix of a square matrix is the matrix of cofactors of each element, i.e.. 

The cofactor of a determinant, which we first defined in Section 3.2, is characterized by a row and column index, in much the same way as we characterize the elements in a matrix. Thus, we can form the matrix of cofactors by accommodating each cofactor in its appropriate position. For example, the determinant ... 

Each signed minor c,y is called a cofactor, and each is an element in an (n x n) matrix of cofactors C. Then the inverse of A can be obtained by... 

Before discussing how to calculate the inverse of a matrix, we will introduce the concept of adjoint of a matrix . The adjoint of a matrix is the matrix of cofactors of the transpose of the matrix. The adjoint of the matrix can be found by using the following steps ... 

Adjugate Matrix of a Matrix Let Ay denote the cofactor of the element Oy in the determinant of the matrix A. The matrix B where B = (Ay) is called the adjugate matrix of A written adj A = B. The elements by are calculated by taking the matrix A, deleting the ith row and Jth. column, and calculating the determinant of the remaining matrix times (—1) Then A" = adj A/lAl. This definition may be used to calculate A"h However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under Numerical Analysis and Approximate Methods. ... 

Note These are examples of important transporters involved in substrate and ADP uptake into the matrix compartment as indicated, and most are reversible. These transporters are proteins and several have been isolated and sequenced. Other specific carriers occur in mitochondria from other tissues. The inner membrane does not allow rapid exchange of NAD or CoA but there are mechanisms for the slow uniport of cofactors synthesized extramitochondrially. 

The dependence (1 TP of v, on ATP is modeled as in the previous section, using an interval C [—00,1] that reflects the dual role of the cofactor ATP as substrate and as inhibitor of the reaction. All other reactions are assumed to follow Michaelis Menten kinetics with ()rs E [0, 1], No further assumption about the detailed functional form of the rate equations is necessary. Given the stoichiometry, the metabolic state and the matrix of saturation parameter, the structural kinetic model is fully defined. An explicit implementation of the model is provided in Ref. [84],... 

In any orbit of the Rydberg electron, most of the time is spent when the electron is far from the nucleus, where the wavefunction is most reasonably characterized in terms of Ah the coefficients of the 0, wavefunctions. Correspondingly, many of the properties depend upon the values of At in a very direct way. As shown by Cooke and Cromer,3 a particularly attractive feature of QDT is that the values of A,2, i.e. the composition of the wavefunction in terms of the collision channels, can be determined by inspecting the quantum defect surface. If we define the cofactor matrix Cia of the matrix of Eq. (20.12) by... 

For the calculation of cofactors we use algorithms based on work by Lowdin [16], and Prosser and Hagstrom [25,26]. An overview of the theory of determinants, cofactors, adjugates and compound matrices can be found in a book by Aitken [17]. The symmetry and possible orthogonality in the orbital spaces give rise to a block-structure in the overlap matrices. This structure is exploited [22,27] to minimise the size of the matrices in the L-d-R decomposition, described below, an n3 process for each matrix. 

Now the adjugate matrix of S, which contains the first order cofactors can be calculated using [17] and Eq. (38)... 

Third, for convenience some reactants are left out of the network. For example, rows are not included in the stoichiometric matrix for cofactor molecules NAD, NADP, and coenzyme A [204], Leaving these cofactors out does not affect the overall flux balance of the reaction network. 

Rischel, C., Spiedel, D., Ridge, J. P., Jones, M. R., Breton, J., Lambry, J. C., Martin, J. L., and Vos, M. H., 1998, Low-frequency vibrational modes in proteins large frequency-shifts induced by point-mutations in ftie protein-cofactor matrix of bacterial reaction centers. Proc. Natl. Acad. Sci. USA, 95 12306nl2311. 

Contrary to the cytochrome P-450 system, most hepatic Phase II enzymes are located in the cytoplasmic matrix. In order for these reactions to occur efficiently, adequate activity of the enzymes involved is essential. In addition, it is clear that adequate intracellular contents of cofactors such as NADPH,... 

The notation u = (u u ), v = (v v ) has been used. These simplifications are easy to introduce in any method based upon the utilization of cofactors. This is true whether the cofactors are computed directly, or indirectly as elements of the adjoint matrix of the overlap —provided only that the block diagonal form of is fully exploited. 

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