Matrices

A *matrix* is defined as a rectangular array of numbers, *e.g.*,

which is a (``two by two'') matrix. A general matrix may be , where is the number of

Either square brackets or large parentheses may be used to delimit the matrix. The th element

The *transpose* of a real matrix
is denoted by
and is defined by

While is , its transpose is . We may say that the ``rows and columns are interchanged'' by the transpose operation, and transposition can be visualized as ``flipping'' the matrix about its main diagonal. For example,

A *complex matrix*
, is simply a
matrix containing complex numbers. The
*transpose* of a complex matrix is normally defined to
include *conjugation*. The conjugating transpose operation is called the
*Hermitian transpose*. To avoid confusion, in this tutorial,
and the word ``transpose'' will always denote transposition
*without* conjugation, while conjugating transposition will be
denoted by
and be called the ``Hermitian transpose'' or the
``conjugate transpose.'' Thus,

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