Matrix algebra equality

The unit matrix (German Einheit) is one which is diagonal with all of the diagonal elements equal to one. It plays the role of unity in matrix algebra. Clearly, the unit matrix multiplied by a constant yields a diagonal matrix with all of die diagonal elements equal to the value of the constant. If the constant is equal to zero, the matrix is the null matrix 0, with all elements equal to zero. 

There is an important theorem in matrix algebra which states that the sum of the eigenvalues of a matrix is equal to the sum of the diagonal matrix elements. Thus... 

It can be proven that a necessary and sufficient condition for there to exist a matrix that will simultaneously diagonalize the Hermitian matrices A and B is for AB to equal BA. To which theorem in Chapter 1 does this matrix-algebra theorem correspond ... 

There are certain rules for adding, subtracting, and dividing matrices these are the rules of matrix algebra. It should be noted first that two matrices are equal only if they are identical. If// =. , then ai = btj for all i and /. 

Strictly speaking, the rules of matrix algebra do not allow us, on the basis of Eqs. 3.2.5 and 2.2.10, to assert that [D] and [B] [r] are equal. The equality of these two matrices is an assumption, albeit the only reasonable way to relate the Fick diffusion coefficients to the Maxwell-Stefan diffusion coefficients >,y. The equality... 

The computation of the fluxes from either of Eqs. 8.3.24 necessarily involves an iterative procedure (except for the special cases discussed above), partly because the themselves are needed for the evaluation of the matrix of correction factors and also because an explicit relation for the matrix [0] cannot be derived as a generalization of Eq. 8.2.16 for binary mass transfer there is no requirement in matrix algebra for the matrices [FFq] be equal to each other even though the fluxes calculated from both parts of these equations must be equal. Indeed, these two matrices will be equal only in the case of vanishingly small mole fraction differences (yg Tg) and vanishingly small mass transfer rates. In almost all cases of interest these two matrices are quite different. An explicit solution was possible for binary systems only because all matrices reduce to scalar quantities. 

Therefore, we see that the boundary conditions at the interface lead to a set of four linear algebraic equations for the constants A, C), B2, D2. The condition for existence of a nontrivial solution of this set of algebraic equations is that the determinant of the coefficient matrix must equal to zero. This condition leads to a complicated algebraic equation relating the dimensionless growth-rate coefficient a to the dimensionless wave number a for specified values of the fluid viscosities, the fluid densities, and the interfacial tension. As usual, stability is determined by the sign of the real part of a. 

Similarly, the number of linearly independent rows of A is called the row-rank of A. The row-rank of A is the column-rank of A. A fundamental theorem in matrix algebra states that the row-rank and the column-rank of a matrix are equal (and equal to the rank) [Schott 1997], Hence, it follows that the rank r(A) < min(/,/). The matrix A has full rank if and only if r(A) = mini/,/). Sometimes the term full column-rank is used. This means that r(A) = min(/,/) =. /, implying that J < I. The term full row-rank is defined analogously. 

Proofs of these equalities can be found in standard matrix algebra books [Magnus Neudecker 1988, Schott 1997, Searle 1982],... 

Matrices and mathematical operators have some things in common. There is a well-defined matrix algebra in which matrices are operated on and this matrix algebra is similar to operator algebra. Two matrices are equal to each other if and only if both have the same number of rows and the same number of columns and if every element of one is equal to the corresponding element of the other. The sum of two matrices is defined by... 

An efficient way to treat such a system is to assemble all coefficients of the different terms of the mass-balance equations in a matrix and to apply methods of matrix algebra to solve the system for steady-state concentrations (level III) or for the concentrations as functions of time (level IV) [19]. We denote the matrix of coefficients (the fate matrix ) by S, the vector of concentrations in all boxes of the model by c, and the vector of all source terms by q. The set of mass-balance equations describing the temporal changes of the concentrations in all boxes then reads c = -S c + q. The steady-state solution is obtained by setting c equal to zero and solving for c. This leads to ss -1. j obtain the steady-state concentrations the emission vector has to be multiplied by the inverse of the matrix S. For the dynamic solutions of the system, the eigenvalues and eigenvectors of S have to be determined. 

If AB = C, then = (AB) = B A In words, the inverse of a product of matrices is equal to the product of inverses, but with the order reversed. We can easily show that this satisfies the mies of matrix algebra. C C = (AB) AB = b- a-iab=b-Mb=b- b=i. If we failed to reverse the order, we would instead have A B AB and, because the matrices do not commute, we would be prevented from carrying through the r uction to 1. 

In matrix multiplication, a matrix times its inverse such as XX equals the identity matrix (ones on the diagonal, zero elsewhere), and in matrix algebra the identity matrix has the same role that the integer one has in algebraic multiplication which means that XX can be inserted or removed at will in a matrix equation. If we insert XX between A and B in Eq. (3.14) and premultiply and postmultiply each side of this equation hy X and X, respectively, we obtain... 

The transpose of Sf in Eq. (14.55) equals the transpose of (US) which equals S U since in matrix algebra the transpose of a product equals the product of the transposes taken in reverse order. Therefore we can write... 

In matrix algebra when two matrices are equal such as A = B, it means that every element of the A matrix is equal to the corresponding element in the B matrix. When matrices are added or subtracted such as A — B it means that every element of the B matrix is added or subtracted from the corresponding element of the A matrix. If a matrix is multiplied by a constant then each element of the matrix is multiplied by that constant. 

Note that the matrix of stoichiometric coefficients devotes a row to each of the N components and a column to each of the M reactions. We require the reactions to be independent. A set of reactions is independent if no member of the set can be obtained by adding or subtracting multiples of the other members. A set will be independent if every reaction contains one species not present in the other reactions. The student of linear algebra will understand that the rank of v must equal M. 

This is where we see the convergence of Statistics and Chemometrics. The cross-product matrix, which appears so often in Chemometric calculations and is so casually used in Chemometrics, thus has a very close and fundamental connection to what is one of the most basic operations of Statistics, much though some Chemometricians try to deny any connection. That relationship is that the sums of squares and cross-products in the (as per the Chemometric development of equation 70-10) cross-product matrix equals the sum of squares of the original data (as per the Statistics of equation 70-20). These relationships are not approximations, and not within statistical variation , but, as we have shown, are mathematically (algebraically) exact quantities. 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

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 ... 

An algebraic criterion for orthogonality is stated in terms of a relation between a matrix and its transpose. The transpose of a matrix A is a matrix A such that the i, element of A is equal to the j, element of A. For example, the... 

The algebra of matrices gives rules for (1) equality, (2) addition and subtraction, (3) multiplication, and (4) division as well as (5) an associative and a distributive law. It also includes definitions of (6) a transpose, adjoint and inverse of a matrix. 

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