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The identity matrix of a given order is the diagonal matrix with all the elements on the main diagonal equal to unity. It is denoted by 1, or by 1„ to indicate the order explicitly.

A matrix with only one column is a column matrix. An «XI column matrix can be identified with a vector in n-dimensional space, and we shall indicate such matrices by boldface letters, as used for vectors.f A matrix with only one row is a row matrix; its transpose is a column matrix, so we denote it as the transpose of a vector. The elements of a row or column matrix will be written with only one subscript; for example,

A set of m nX 1 vectors /'= 1,2 m, is linearly independent if and only if the only coefficients a,, /= l,2,...,m, satisfying the equation

2 a,B(,) = a,B(,) + a2B<2)+ • • • + amB("> = 0

'The word adjoint is sometimes used for a different matrix in the literature.

tStrictly speaking, a vector is an abstract mathematical object, and the column matrix is a concrete realization of it, the matrix elements being the components of the vector in some coordinate system.

are a, = 0, i= 1,2, m. There can never be more than n linearly independent n X 1 vectors.

C.2 Matrix Algebra

Multiplication of a matrix by a scalar is accomplished by multiplying each element of the matrix by the scalar, i.e., sA=[sAv] (C-10)

Addition of two matrices is possible only if the matrices have the same order. The elements of the matrix sum are the sums of the corresponding elements of the matrix addends, i.e.,

Matrix subtraction follows from the above two rules by

Multiplication of two matrices is possible only if the number of columns of the matrix on the left side of the product is equal to the number of rows of the matrix on the right. If A is of order IXm and B is mX n, the product AB is the IXn matrix given by

Matrix multiplication is associative

and distributive over addition

A(B+C) = AB + AC (C-15) but is not commutative, in general,

In fact, the products AB and BA are both defined and have the same order only if A and B are square matrices, and even in this case the products are not necessarily equal. For the square matrices ^=[34] and for example, we have

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