Matrix algebra subtraction

Matrix algebra also involves the addition and subtraction of matrices. The rules for this are as follows ... 

An algebra typically involves the operations of adding, subtracting, multiplying, or dividing the objects it describes, whether matrices or simple numbers. For completeness, we now summarize some other aspects of matrix algebra, built on the fundamental definitions of addition/subtraction (9.8), scalar multiplication (9.9), and matrix multiplication (9.11). 

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 /. 

Matrix algebra provides a powerful method for the manipulation of sets of numbers. Many mathematical operations — addition, subtraction, multiplication, division, etc. — have their counterparts in matrix algebra. Our discussion will be Umited to the manipulations of square matrices. For purposes of illustration, two 3x3 matrices will be defined, namely... 

Be comfortable working with basic operations of matrix algebra (addition, subtraction, multiplication)... 

The algebra of multi-way arrays is described in a field of mathematics called tensor analysis, which is an extension and generalization of matrix algebra. A zero-order tensor is a scalar a first-order tensor is a vector a second-order tensor is a matrix a third-order tensor is a three-way array a fourth-order tensor is a four-way array and so on. The notions of addition, subtraction and multiplication of matrices can be generalized to multi-way arrays. This is shown in the following sections [Borisenko Tarapov 1968, Budiansky 1974],... 

Just as in scalar operations, where we have addition, subtraction, multiplication and division, we also have addition, subtraction, multiplication and inverse (playing the role of division) on matrices, but there are a few restrictions in matrix algebra before these operations can be carried out. 

As in the case of scalar quantities, matrices can also be added, subtracted, multiplied and so forth, but special rules of matrix algebra must be followed. 

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. 

The normal rules of association and commutation apply to addition and subhaction of matrices just as they apply to the algebra of numbers. The zero matrix has zero as all its elements hence addition to or subtraction from A leaves A unchanged... 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

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. 

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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