Algebra symmetric matrix

Equation (31.3) defines the eigenvalue decomposition (EVD), also referred to as spectral decomposition, of a square symmetric matrix. The orthonormal matrices U and V are the same as those defined above with SVD, apart from the algebraic sign of the columns. As pointed out already in Section 17.6.1, the diagonal matrix can be derived from A simply by squaring the elements on the main diagonal of A. 

The Hessian matrix H(r) is defined as the symmetric matrix of the nine second derivatives 82p/8xt dxj. The eigenvectors of H(r), obtained by diagonalization of the matrix, are the principal axes of the curvature at r. The rank w of the curvature at a critical point is equal to the number of nonzero eigenvalues the signature o is the algebraic sum of the signs of the eigenvalues. The critical point is classified as (w, cr). There are four possible types of critical points in a three-dimensional scalar distribution ... 

Sylow, 6. symbol list, xv. symmetric matrix, 59. symmetry element, 7. symmetry operations, 7, 10 algebra of, 15 for symmetric tripod, 27. symmetry operator, 10. symmetry orbitals, 203, 206, 207, 210, 212, 246. 

The representation of an EMB(A) by its fee-matrix B = (b ), an n x n symmetric matrix with positive integer entries and well-defined algebraic properties, corresponds to the act of translating an object of chemistry into a genuine mathematical object. This implies that the chemistry of an EMB(A) corresponds to the algebraic properties of the matrix B. 

Indeed, immediately following Eq. (10.2.3) it was assumed that the matrix K for an n-component system possessed i characteristic nonpositive roots. This assumption is justified by the fact that the similarity transformation gives a symmetric matrix R, as was shown in Section 10.3 to be the case as a result of the principle of microscopic reversibility. But it is known from matrix algebra that a symmetric (n X n) matrix possesses n characteristic nonpositive real roots. Since, furthermore, R has the same roots as K, as was mentioned in Section 10.3, the assumption made is justified. Therefore any composition vector a will go to its equilibrium value without oscillating about the latter, as shown clearly by Eq. (10.2.23). 

Q j = -f) where is a unit matrix of order n. From linear algebra, it is known that if 0 is an arbitrary nondegenerate skew-symmetric matrix, it can be reduced to the above-mentioned canonical form through a linear nondegenerate transformation of the basis. 

High-order models are often a result of models consisting of many differential equations or partial differential equations that have been converted into ordinary differential equations. These types of model are adequate for simulations studies but are not suitable for online use. A popular technique of model reduction that does not make use of error minimization is the model balancing method. The procedure is to find observability and controllability Gramians so as to determine which states have the largest overall contribution to the model. In systems theory and linear algebra, a Gramian matrix is a real-values symmetric matrix that can be used for a test for linear independence of functions. A system is called controllable if all states X can be influenced by the control input vector w, a system is observable if all states can be determined from the measurement vector jp. 

To remove the rotation, we can multiply F by its transpose (interchanging rows and columns). We know ftom matrix algebra that if we multiply a matrix times its transpose, we always get a symmetric matrix. Recall that transpose simply means interchanging rows and columns of the matrix. 

The above constitute a set of simultaneous linear algebraic equations, called the normal equations. The matrix X X) is a (/ x k) symmetric matrix. Assumption 4 made earlier guarantees that (X X) is nonsingular therefore, its inverse exists. Thus, the normal equations can be solved for the vector b ... 

It is a general result of linear algebra that the eigenvalues of a symmetric matrix A are unaffected by similarity transformations TAT [10]. From the orthonormality relation (10.7.2), we find that the inverse of C is given as... 

The matrix gab is symmetric and non-singular and has other purely algebraic properties that can be demonstrated by first defining the matrices... 

A matrix basis for group algebras of symmetric groups... 

Just as we saw with the symmetric groups, groups of spatial operations have associated group algebras with a matrix basis for this algebra,... 

Note that and A r need not be the same matrix (although they are for the important special case of real symmetric A that we are most concerned with). Note also that AT1 need not exist, even if A 0. A matrix for which A"1 exists is called nonsingular (see below) and leads to many arithmetic extensions that are not permitted to singular matrices. The many varieties of singularity (not just A = 0) and the (potentially) noncommutative aspects of multiplication distinguish matrix algebra from its scalar counterpart in interesting ways. 

If W is a finite matrix, linear algebra tells us that this can in fact be asserted when W is symmetric. If W is an operator in an infinite-dimensional space the mathematical conditions are considerably more complicated, but as a rule of thumb one may also regard any symmetrical operator as diagonaliz-able - as is customary in quantum mechanics. It will now be shown that the detailed balance property guarantees that the operator W is symmetrical. We adopt the notation of a continuous range. 

Equations (723) and (727) therefore represent a closed, cyclically symmetric, algebra in which all three space-like components are meaningful. The cyclical commutator basis can be used to build a matrix representation of the three spacelike magnetic components of the electromagnetic wave in the vacuum... 

Therefore, symmetrical transformations in the crystal are formalized as algebraic (matrix-vector) operations - an extremely important feature used in all crystallographic calculations in computer software. The partial list of symmetry elements along with the corresponding augmented matrices that are used to represent symmetry operations included in each symmetry element is provided in Table 1.19 and Table 1.20. For a complete list, consult the Intemational Tables for Crystallography, vol. A. 

As a prelude to the proof as such it will be found convenient to cast the Huckel equations in matrix form as we have continually emphasised, much of Huckel theory is essentially the linear algebra associated with real-symmetric matrices, albeit in a somewhat disguised form. Let us begin by considering again (in the original matrix-notation rather than the a, / notation we have lately adopted) the rth secular equation, satisfied by e, and c/r, r = 1,2,..., n. This is... 

From the onset, the three-dimensional algebraic model presents a consistent treatment of this problem. Let us recall the Hamiltonian matrix for the first polyad of a linear triatomic molecules [Eq. (4.58)], written here in the symmetrical case, A = A = A2 and N = = N2 ... 

A general theorem in linear algebra, see Ref. [10] for more details, says that any finite matrix representation can be transformed to complex symmetric form. For this reason, we can without loss of generality set up a complex symmetric model for our generalized dynamical picture based entirely on the parameters //u, H22, and v with... 

Higham, N. J., Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Applic., 103, 103 (1988). 

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