Square matrix determinant

In order to determine the matrix thresholds, we present an expression of the coefficients dispersion that is related to the flattening of the cloud of the points around the central axis of inertia. The aim is to measure the distance to the G barycentre in block 3. So, we define this measure Square of Mean Distance to the center of Gravity as follow ... 

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. 

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 determinant of a square matrix of order two is called a determinant of order two and is defined as... 

The determinant of a square matrix C (det C) is defined as the sum of all possible products found by taking one element from each row in order from the top and one element from each column, the sign of each product multiplied by (-ly, where r is the number of times the column index decreases in the product. 

EXPLICIT EXPRESSION OF THE DETERMINANT OF AN ARBITRARY SQUARE MATRIX... 

Using the NSS, one can reformulate the expression which gives the determinant of an arbitrary (nxn) square matrix A, Det A. A compact formula of Det I AI can be written in this way as ... 

By way of example we construct a positive semi-definite matrix A of dimensions 2x2 from which we propose to determine the characteristic roots. The square matrix A is derived as the product of a rectangular matrix X with its transpose in order to ensure symmetry and positive semi-definitiveness ... 

The major problem is to find the rotation/reflection which gives the best match between the two centered configurations. Mathematically, rotations and reflections are both described by orthogonal transformations (see Section 29.8). These are linear transformations with an orthonormal matrix (see Section 29.4), i.e. a square matrix R satisfying = RR = I, or R = R" . When its determinant is positive R represents a pure rotation, when the determinant is negative R also involves a reflection. 

For a square matrix A, there exists a number called the determinant of the matrix. This determinant is denoted by... 

For most students, their first encounter with matrices is in the study of determinants. However, a determinant is a very special case in winch a given square matrix has a specific numerical value. If the matrix A is of order two, its determinant can be written in the form... 

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. 

To follow this procedure it is useful to define the determinant associated with the square matrix of interest, written as... 

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. 

As mentioned earlier, singular matrices have a determinant of zero value. This outcome occurs when a row or column contains all zeros or when a row (or column) in the matrix is linearly dependent on one or more of the other rows (or columns). It can be shown that for a square matrix, row dependence implies column dependence. By definition the columns of A, a, are linearly independent if... 

The determinant det A of the square matrix A xn is most simply defined by the row-wise recurrence formula... 

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. 

Other notation used diagB is the diagonal n x n matrix consisting of the diagonal elements of the square matrix B. The trace of B is denoted trB, and the determinant of B is denoted B. The Kronecker product of two matrices is denoted by symbol (g). Other notation will be introduced as needed. 

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... 

Candidate mineral compositions or test vectors are tested by linearly rotating the NC eigenvectors towards a test vector by using a least squares procedure ( ) and determining if the test vector could possibly lie in the vector space defined by the NC eigenvectors. In this way, suspected minerals are kept or rejected from further consideration. (From this step of the analysis, TTFA derives its name.) When NC mineral compositions have been determined that adequately reproduce the original data and are consistent with other information, such as XRD or Infrared analysis, this aspect of TTFA is finished. At this point, we have successfully determined the matrix C of Equation 5. 

Francl et al. (1996) examined the conditioning of the least squares matrix in the fitting procedure, and conclude that the method cannot be used to assign statistically valid charges to all atoms in a given molecule. This problem cannot be alleviated by the selection of more sampling points, and thus may require the introduction of chemical constraints to reduce the number of charges to be determined. 

Analogous but more complicated formulas define the determinants of square matrices of higher dimensions. (A square matrix is a matrix which has the same number of rows as columns.) It is not possible to define the determinant of a non-square matrix. 

Two important complex numbers associated to any particular complex linear operator T (on a finite-dimensional complex vector space) are the trace and the determinant. These have algebraic definitions in terms of the entries of the matrix of T in any basis however, the values calculated will be the same no matter which basis one chooses to calculate them in. We define the trace of a square matrix A to be the sum of its diagonal entries ... 

The determinant of a square matrix is the determinant obtained by considering the array of elements in the matrix as a determinant if the matrix is A we will write the determinant as det(A) i.e. if... 

With any square matrix A, we can associate a determinant, symbolized by detA. Obviously detl= 1. A useful theorem is... 

The functions (2.50) are called basis functions The matrices F, G,. .. are called matrix representatives of the operators F, G,. .. in the

specific form of the matrix representation of a set of operators depends on the basis chosen. Equation (2.53) shows that the effect of the operator G on the basis functions is determined by the matrix elements GkJ. Since an arbitrary well-behaved function can be expanded using the complete set (2.50), knowledge of the matrix G allows one to determine the effect of the operator G on an arbitrary function. Thus, knowledge of the square matrix G is fully equivalent to knowledge of the corresponding operator G. Since G is a Hermitian operator, its matrix elements satisfy Gij = (GJi). Hence the matrix G representing G is a Hermitian matrix (Section 2.1). 

Let us emphasize that we have made no approximations yet. Equation (3.13) is a set of simultaneous differential equations for the coefficients cm that determine the state function (3.13) is fully equivalent to the time-dependent Schrodinger equation. [The column vector c(/) whose elements are the coefficients ck in (3.8) is the state vector in the representation that uses the tyj s as basis functions. Thus (3.13) is a matrix formulation of the time-dependent Schrodinger equation and can be written as the matrix equation ihdc/dt = Gc, where dc/dt has elements dcmf dt and G is the square matrix with elements exp(.iu>mkt)H mk. ... 

This secular equation is a polynomial in A of degree equal to the order of the square matrix A. Note from Theorem 6 on determinants in Section 1.2 and (2.39) that for a diagonal matrix, the eigenvalues are equal to the diagonal elements. 

Each square matrix is assumed to correspond to a certain value (to be more precise, to a numerical function) which is called a matrix determinant. For a first-order matrix, i.e. the number of atJ, the determinant is equal to this number itself... 

Note that a characteristic polynomial of the square matrix A = o of the order n is called a determinant for a set of linear homogeneous equations... 

R is a column vector with N elements R, r is a column vector with N elements rt, and A is an A x A square matrix with constant elements Ai j. The new momenta Pj may be determined using the definition in Eq. (4.61). From Eq. (D.l) we obtain... 

Determinants. To each square matrix a of dimension n we can associate a determinant det a as follows ... 

If the mattix I) is a square matrix, the estimated values of y are identical with the observed values y. The model provides an exact fit to die data, and there are no degrees of freedom remaining to determine die lack-of-fit. Under such circumstances diere will not be any replicate information but, nevertheless, the values of b can provide valuable information about the size of different effects. Such a situation might occur, for example, in factorial designs (Section 2.3). The residual error between die observed and fitted data will be zero. This does not imply that the predicted model exactly represents die underlying data, simply that the number of degrees of freedom is insufficient for determination of prediction errors. In all other circumstances there is likely to be an error as die predicted and observed response will differ. 

The design matrix is given in Table 2.35, and being a square matrix, the terms can easily be determined using die inverse. For interested readers, the relationship between die two types of models is explored in more detail in die Problems, but in most cases we recommend using a Sheffe model. 

The square matrix T with elements T(ij)(rs) has m(m — 1) rows (in accordance with the number of ordered pairs (ij) or parameters ay) and determines the parameter sensitivity of the azeotrope value towards the accuracy estimation of the reactivity ratios. Really, when their errors are the same, the deviation 8x 2 (4.18) of the theoretically predicted location of azeotrope will more or less depend on the values of elements of matrix T. The calculation of such elements have no principal difficulties since an explicit dependence of x on parameters ay is known. In the case of the rather strong parameter sensitivity, when derivatives of xf with the respect to ay are large, even comparatively small errors in 8ay may result in substantial errors in calculations of x making it quite impossible to predict theoretically the existence or absence of an azeotrope in the given system. The examples of such systems were discussed earlier [125, 132, 135, 139] but as far as the author knows nobody has yet carried out the quantitative consideration of the parameter sensitivity by means of the expressions (4.18). 

The real parts of the eigenvalues are negative, and the perturbations will decay in time as Figure 12.3 illustrates. When the value of B is 2.4 then the oscillations are sustainable. Figure 12.3b and 12.3d show the state-space plot of concentrations Cx and CY for the different values of B. Regardless of whether the eigenvalues are real or complex, the steady state is stable to small perturbations if the two conditions tr[J] < 0 and dct. / > 0 are satisfied simultaneously. Here, tr is the trace and det is the determinant of the square matrix J. 

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