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.
For example, the general matrix is
(Either square brackets or large parentheses may be used.) Theth element7.3 of a
matrix may be denoted by or . The rows and
columns of matrices are normally numbered from instead of from; thus, and . When , the
matrix is said to be square.
The transpose of a real matrix
is denoted by and is defined by
Note that while is , its transpose is .
A complex matrix, is simply a
matrix containing complex numbers. The transpose of a complex matrix is normally defined to includeconjugation. The conjugating transpose operation is called theHermitian transpose. To avoid confusion, in this tutorial,
and the word ''transpose'' will always denote transpositionwithout conjugation, while conjugating transposition will be
denoted by and be called the ''Hermitian transpose'' or the
''conjugate transpose.'' Thus,
Example: The transpose of the general matrix is
while the conjugate transpose of the general matrix is
A column-vector
is the special case of an matrix, and a row-vector
(as we have been using) is a matrix. In contexts where
matrices are being used (only this section for this course), it is best to
define all vectors as column vectors and to indicate row vectors
using the transpose notation, as was done in the equation above.
(''two by two'') matrix. A general matrix may be
, where
is the number of rows, and
is the number of columns. For example, the general
matrix is
th element7.3 of a matrix
may be denoted by
or
. The rows and columns of matrices are normally numbered from
instead of from
; thus,
and
. When
, the matrix is said to be square.
is denoted by
and is defined by
.
, is simply a matrix containing complex numbers. The transpose of a complex matrix is normally defined to includeconjugation. The conjugating transpose operation is called theHermitian transpose. To avoid confusion, in this tutorial,
and be called the ''Hermitian transpose'' or the ''conjugate transpose.'' Thus,
matrix, and a row-vector
matrix. In contexts where matrices are being used (only this section for this course), it is best to define all vectors as column vectors and to indicate row vectors using the transpose notation, as was done in the equation above.
