The Transpose(A) function computes the transpose of A.

If T := Transpose(A), then one of the following conditions holds:

if A is a Matrix, then T is a Matrix with $T_{i,j}=A_{j,i}$;

if A is a Vector, then T is a Vector containing the same elements as A, and opposite in orientation to A; or

if A is a scalar value, then $T=A$.

The $\mathrm{HermitianTranspose}\left(A\right)$ function computes the Hermitian transpose (dual) of A.

If H := HermitianTranspose(A), then one of the following conditions holds:

if A is a Matrix, then H is a Matrix with $H_{i,j}=\stackrel{\&conjugate0;}{A_{j,i}}$;

if A is a Vector, then H is a Vector containing the conjugated elements of A, and opposite in orientation to A; or

if A is a scalar value, then $H=\stackrel{\&conjugate0;}{A}$.

$\mathrm{Transpose}\left(A\right)$ can also be entered as $A^{+}$ or as $A^{\mathrm{\%T}}$.  $\mathrm{HermitianTranspose}\left(A\right)$ can also be entered as $A^{*}$ or as $A^{\mathrm{\%H}}$.  These short forms do not require the LinearAlgebra package to be loaded first.

The inplace option (ip) determines where the result is returned. If given as inplace=true, the result overwrites the first argument. If given as inplace=false, or if this option is not included in the calling sequence, the result is returned in a new Matrix (or Vector).

The condition inplace=true can be abbreviated to inplace.

The inplace option must be used with caution since, if the operation fails, the original Matrix (or Vector) argument may be corrupted.

Note: For this function, the inplace option is only valid for input that is a Vector or a square Matrix.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix or Vector constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

The inplace and constructor options are mutually exclusive.

These functions are part of the LinearAlgebra package, and so they can be used in the form Transpose(..) or HermitianTranspose(..) only after executing the command with(LinearAlgebra). However, they can always be accessed through the long form of the command by using LinearAlgebra[Transpose](..) or LinearAlgebra[HermitianTranspose](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$A≔⟨⟨1\,2\,w⟩\left|⟨2\,6\,x⟩\right|⟨3\,7\,y⟩|⟨4\,8\,z⟩⟩$

$A≔\left[\begin{array}{cccc}1& 2& 3& 4\\ 2& 6& 7& 8\\ w& x& y& z\end{array}\right]$

$\mathrm{Transpose}\left(A\right)$

$\left[\begin{array}{ccc}1& 2& w\\ 2& 6& x\\ 3& 7& y\\ 4& 8& z\end{array}\right]$

$A^{\mathrm{\%T}}$

$\left[\begin{array}{ccc}1& 2& w\\ 2& 6& x\\ 3& 7& y\\ 4& 8& z\end{array}\right]$

$V≔⟨a\,b\,c⟩$

$V≔\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$

$\mathrm{HermitianTranspose}\left(V\,\mathrm{inplace}=\mathrm{true}\right)\:$

$V$

$\left[\begin{array}{ccc}\stackrel{\&conjugate0;}{a}& \stackrel{\&conjugate0;}{b}& \stackrel{\&conjugate0;}{c}\end{array}\right]$

$V^{\mathrm{\%H}}$

$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$

$\mathrm{HermitianTranspose}\left(x\right)$

$\stackrel{\&conjugate0;}{x}$