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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 rows,
and
is the number of columns of the matrix.
For example, the general
matrix is
Either square brackets or large parentheses may be used to delimit the
matrix. The
th elementH.1 of a matrix
may be denoted by
,
, or
. For example,
in the
above two examples. The rows and columns of matrices are normally
numbered from
instead of from 0
; thus,
and
. When
, the matrix is said to be
square.
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,
Subsections
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