Matrices multiplication rule

The operator q2 is the square of the operator q. Since matrix representatives of operators obey the same relations as the operators (Section 2.3), the q2 matrix [whose elements are ] is the square of the q matrix [whose elements are (4.45)]. Hence, using the matrix-multiplication rule (2.11), we have... 

We recognize (2.54) as the matrix-multiplication rule (2.11). Hence the matrix A is the product of the matrices F and G A = FG, the same relation obeyed by the corresponding operators. Actually, we proved this earlier— see (1.52). 

To verify the matrix equation (8.88), we show that each element (HC),y of the matrix HC equals the corresponding element (CW),j of CW. TTie matrix-multiplication rule (7.106) gives (HC)y = kfliki kj Consider the eigenvalue equation = W c ... 

From the matrix multiplication rule, the left side of (13.178) is the (r, j)th element of FC, and the right side is the (r, i)th element of S(Ce). Since the general element of FC equals the general element of SCe, these matrices are equal ... 

Using matrix multiplication rules, the Dirac equation [Eq. (3.54)] with bispinors can be rewritten in the form of two equations with spinors and 0 ... 

Use of the matrix multiplication rule (7.107) gives the (, i)th element of the matrix product Ce as (Ce)s, = LmCsmSmi = = CjiS . Hcuce the Roothaan equa-... 

Using the matrix multiplication rule for Kronecker products... 

The matrix equation [E.l] involves the multiplication of the matrices [A] and (x). To do this one must apply the simple rules of matrix multiplication. These are ... 

Using the rules of matrix multiplication it is further shown that these matrices reproduce the C2v multiplication table shown in Table 2 and readily obtained from the diagram below. As an example, from the table it follows that avC2 = cr v as shown by the matrix multiplication. 

The matrix exponential eAl is calculated using the standard rules of matrix multiplication... 

Working with Markov chains, confusion is bound to arise if the indices of the Markov matrix are handled without care. As stated lucidly in an excellent elementary textbook devoted to finite mathematics,24 transition probability matrices must obey the constraints of a stochastic matrix. Namely that they have to be square, each element has to be non-negative, and the sum of each column must be unity. In this respect, and in order to conform with standard rules vector-matrix multiplication, it is preferable to interpret the probability / , as the probability of transition from state. v, to state s (this interpretation stipulates the standard Pp format instead of the pTP format, the latter convenient for the alternative 5 —> Sjinterpretation in defining p ), 5,6... 

Vectors have earlier been introduced as matrices with one dimension being reduced to one, i.e. they are comprised of one column or one row only. When the rule for matrix multiplication given by equation (2.6) is formally applied to vectors and the appropriate dimensions match, there are two immediate consequences ... 

According to the rule for matrix multiplication introduced earlier, each element of y is calculated as the scalar product between c and the corresponding column of A. These linear operations are represented exactly by the following system of inhomogeneous linear equations ... 

Equations (4.11) and (4.26) have to be identical. Simple verification based on the rules for matrix multiplication ... 

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