Multiplication of Two Matrices

Many of the important uses of matrices depend on the definition of multiplication. Multiplication is defined only for conformable matrices. Two matrices [A] and [B] are said to be conformable in the order [y4][B] if [ ] has the same number of columns as [B] has rows. 

For example, two square matrices of order 2, [ 4] and [B], when multiplied together give a third matrix [C] 

A square matrix [ ] of order 2 and a column matrix (h) of order 2 when multiplied together 

A row matrix of order m can be multiplied by a column matrix of order m in two different ways. The inner product is a scalar defined by 

The outer product of a row and column matrix, both of order m, is a square matrix of order m and is written as [C] = with 

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Multiplication of two matrices is the most important operation in multivariate data analysis. It is not performed element-wise, but each element of the resulting matrix is a scalar product (see Section A.2.3). A matrix A and a matrix B can be multiplied by A B only if the number of columns in A is equal to the number of rows in B this... 

This is the same rule as the row by column rule for the multiplication of two matrices. 

Multiplication of two matrices can be either scalar multiplication or vector multiplication. Scalar multiplication of two matrices consists of multiplying corresponding elements, i.e.. 

The vector multiplication of two matrices is somewhat more complicated ... 

Vector multiplication of two matrices is possible only if the matrices are conformable, that is, if the number of columns of A is equal to the number of rows of B. The opposite condition, if the number of rows of A is equal to the number of columns of B, is not equivalent. The following examples, involving multiplication of a matrix and a vector, illustrate the possibilities ... 

A matrix is a list of quantities, arranged in rows and columns. Matrix algebra is similar to operator algebra in that multiplication of two matrices is not necessarily commutative. The inverse of a matrix is similar to the inverse of an operator. If A is the inverse of A, then A A = AA = E, where E is the identity matrix. We presented the Gauss-Jordan method for obtaining the inverse of a nonsingular square matrix. 

The multiplication of two matrices is defined as where an element of the j C matrix is obtained by... 

Fig. 2.1 Matrix representation of a group the operators (left) are mapped onto the transformations (right) of a function space. The consecutive action of two operators on the left (symbolized by ) is replaced by the multiplication of two matrices on the right (symbolized by x)...

Chapter 9 dealt with the basic operations of addition of two matrices with the same dimensions, of scalar multiplication of a matrix with a constant, and of arithmetic multiplication element-by-element of two matrices with the same... 

The trace of a product of two matrices, which may or may not commute, is independent of the order of multiplication... 

Hie multiplication of matrices requires a bit more reflection. The product C of two matrices A and B is usually defined by C = AB if... 

While the matrix multiplication defined by Eq. (28) is the more usual one in matrix algebra, there is another way of taking the product of two matrices. It is known as the direct product and is written here as A <8> 1 . If A is a square matrix of order n and B is a square matrix of order m, then A<8>B is a square matrix of order tun. Its elements consist of all possible pairs of elements, one each from A and B, viz. 

If two matrices are square, they can be multiplied together in any order. In general, the multiplication is not commutative. That is AB BA, except in some special cases. It is said that the matrices do not commute, and this is the property of major importance in quantum mechanics, where it is common practice to define the commutator of two matrices as... 

The product of two matrices AB exists if and only if the number of rows in the second matrix B is the same as the number of columns in the first matrix A. If this is the case, the two matrices are said to be conformable for multiplication. If A is an mxp matrix and B is a pxn matrix, then the product C is an mxn matrix ... 

Matrices such as these for which all elements are zero except for clusters, or blocks, of elements around the diagonal, are said to be in block form. In a set of matrices, all in block form with the same distribution of blocks of non-zero elements, the blocks can be thought of as independent submatrices. Multiplication of two such matrices preserves the block form, and the value of any element in a block of the product depends only on the elements in the corresponding blocks of the factors. 

The possible outcomes of measurements—combinations of scattering matrix elements—listed in Table 13.1 follow from multiplication of three matrices those representing the polarizer, the scattering medium, and the analyzer. If U is an element in the optical train, then the measured irradiance depends on only two matrix elements. In general, however, there are four elements in a combination, so that four measurements are required to obtain one matrix element. 

Addition of two square matrices P + R is performed by taking the sums of corresponding elements, Pmn + Rmn. Multiplication of two square matrices P and R to give a square matrix Q is defined by... 

Now, the multiplication of two square matrices of order m involves multiplication operations and — 1) additions. However, the multiplication of a square and a column... 

Note Multiplication of two square matrices [a ] and [b J, where and are the elements of each matrix, can be carried out as follows ... 

It was remarked by Born and Jordan that this rule for multiplication is identical with one which has long been known in mathematics as the rule for forming the j)roduct of two matrices , such as occur in the theory of linear transformations and the theory of determinants. Wo may thxu eforo regai-d Heisenl)erg s square arrays as infinite matrices, and calculate with them by the known rules of the theory of matrices. 

Determining the inverse of a matrix by hand is a fairly complicated matter. Fortunately, Excel has a built-in function, MINVERSE, that will perform the inversion. It also has a matrix multiplication function, MMULT, that will calculate the product of two matrices. In order to let the spreadsheet know that your instructions concern an entire block or array rather than an individual cell, these two functions require that you first highlight the entire block to which the instruction applies, and then enter the instruction while simultaneously depressing Ctrl, Shift, and Enter. 

Throughout the following development, a vector will always be represented by a column vector, and the transpose of a vector will always be represented by a row vector. The transpose of a product of two matrices (which are conformable in multiplication) is equal to the product of the transposes taken in reverse order... 

The determinant of A is commonly denoted by A or det A. If A = 0, the matrix A is said to be singular and if A 0, the matrix A is said to be nonsingular. One rule for the multiplication of two determinants of the same order is the same rule as that for the multiplication of two square matrices of the same order. Thus... 

The matrix multiplication of two phase matrices requires the Fourier series expansion of the azimuthally dependent functions, so that each term in the Fourier series expansion can be treated independently [48, 49],... 

If a structure is invariant with respect to two symmetry operations (P, tp) and (Q, (q), then it is clearly invariant with respect to the successive application of the two operations. We call this successive application the product. If we first apply (P, tp) and then (Q, Iq), the vector x is transformed into x" = Qx + Iq = QPx + Qtp + tq. The multiplication of the matrices Q and P is thus carried out from right to left. We first apply P and then Q ... 

Multiplication The product AB of two matrices is defined whenever the number of columns of the first matrix is the same as the number of rows of the second matrix. When this condition exists, the matrices are said to be conformable. If A is n x m and B is n x r, then the product AB = C is an OT X r matrix. The element in the rth row and /th colunm of C is obtained by multiplying each element of the rth row of A by the corresponding element of the /th column of B, and then adding the resulting products. That is. 

For this reason, it is necessary to use terminology which specifies the order of multiplication. For example, if the matrices A and B are n x n, then C = AB is read as B premultiplied by A whereas D = BA is read as B postmultiplied by A. Fiuther, the results AB and BA are not necessarily equal. This is a very different phenomenon from the usual symbolic algebra and may even result in the product of two matrices being the null matrix without either matrix being nuU. For instance. 

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