The product of an n X m matrix and an m-dimensional vector (an m X1 matrix) is an «-dimensional vector; thus,

A similar result holds if a row matrix is multiplied on the right by a matrix,

An important special case of the above is the multiplication of a IXn row matrix (on the left) by an n X1 column matrix (on the right) which yields a scalar, r = YTX= 2 XjYJ J" '

For real vectors, this scalar is the inner product, or dot product, or scalar product of the vectors X and Y. For vectors with complex components, it is more convenient to define the inner product by using the adjoint of the left-hand vector rather than the transpose. Thus, in general,

Note that, in general,

This definition reduces to the usual definition for real vectors, for which the inner product is independent of the order in which the vectors appear. Two vectors are orthogonal if their inner product is zero. The inner product of a vector with itself

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