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Transpose

Transpose[list]

transposes the first two levels in list.

Transpose[list,{n₁,n₂,…}]

transposes list so that the k level in list is the n_k level in the result.

Transpose[list,mn]

transposes levels m and n in list, leaving all other levels unchanged.

Transpose[list,k]

cycles the levels in list k positions to the right.

Details and Options

Transpose[m] gives the usual transpose of a matrix m.
Transpose[m] can be input as m^.
 can be entered as tr or \[Transpose].
Transpose[m] formats as in StandardForm and TraditionalForm. »
For a matrix m, Transpose[m] is equivalent to Transpose[m,{2,1}].
For an array a of depth r≥3, Transpose[a] is equivalent to Transpose[a,{2,1,3,…,r}], only transposing the first two levels. »
The n_i in Transpose[a,{n₁,n₂,…}] or Transpose[a,n₁n₂] must be positive integers no larger than ArrayDepth[a].
If {n₁,n₂,…} is a permutation list, then the element at position {i₁,i₂,…} of Transpose[a,{n₁,n₂,…}] is the element at position {i_n₁,i_n₂,…} of the array a.
For a permutation perm, the dimensions of Transpose[a,perm] are Permute[Dimensions[a],perm].
A permutation list perm in Transpose[a,perm] can also be given in Cycles form, as returned by PermutationCycles[perm]. »
Transpose[a,m↔n] or Transpose[a,TwoWayRule[m,n]] is equivalent to Transpose[a,Cycles[{{m,n}}]]. »
Transpose[a,k] is equivalent to Transpose[a,RotateLeft[Range[n],k]], where n is the depth of a.
Transpose allows the n_i to be repeated, computing diagonals of the subarrays determined by the repeated levels. The result is therefore an array of smaller depth.
For a square matrix m, Transpose[m,{1,1}] returns the main diagonal of m, as given by Diagonal[m]. »
In general, if n_p=n_q then the operation Transpose[a,{n₁,n₂,…}] is possible for an array a of dimensions {d₁,d₂,…} if d_p=d_q.
Transpose works on SparseArray and structured array objects.

Examples

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Basic Examples (4)

Transpose a 3×3 numerical matrix:

Visualize the transposition operation:

Transpose a 2×3 symbolic matrix:

Use followed by tr to enter the transposition operator:

Output formatting:

Scope (12)

Matrices (5)

Enter a matrix as a grid:

Transpose the matrix and format the result:

Transpose a row matrix into a column matrix:

Format the input and output:

Transpose the column matrix back into a row matrix:

Transposition of a vector leaves it unchanged:

Transpose leaves the identity matrix unchanged:

s is a sparse matrix:

Transpose[s] is also sparse:

The indices have, in effect, just been reversed:

Transpose a SymmetrizedArray object:

The result equals the negative of the original array, due to its antisymmetry:

Arrays (7)

Transpose the first two levels of a rank-3 array, effectively transposing it as a matrix of vectors:

Transpose an array of depth 3 using different permutations:

Cycle levels of a depth-5 array two positions to the right:

Perform transpositions using TwoWayRule notation:

Perform transpositions using Cycles notation:

Transpose levels 2 and 3 of a depth-4 array:

The second and third dimensions have been exchanged:

Get the leading diagonal by transposing two identical levels:

Applications (13)

Matrix Decompositions (4)

is a random real matrix:

Find the QRDecomposition of :

is orthogonal, so its inverse is :

Reconstruct from the decomposition:

Compute the SchurDecomposition of a matrix :

The matrix is orthogonal, so its inverse is :

Reconstruct from the decomposition:

Compute the SingularValueDecomposition of a matrix :

The matrices and are orthogonal, so their inverses are their transposes:

Reconstruct from the decomposition:

Construct the singular value decomposition of , a random matrix:

First compute the eigensystem of :

The singular values are the square roots of the nonzero eigenvalues:

The matrix is a diagonal matrix of singular values with the same shape as :

The matrix has the eigenvectors as its columns:

The matrix has columns of the form for each of the nonzero eigenvalues:

Verify that and are orthogonal:

Verify the decomposition:

Special Matrices (6)

A symmetric matrix obeys , an antisymmetric matrix . This matrix is symmetric:

Confirm with SymmetricMatrixQ:

This matrix is antisymmetric:

Confirm with AntisymmetricMatrixQ:

A matrix is orthogonal if . Check if the matrix is orthogonal:

Confirm that it is orthogonal using OrthogonalMatrixQ:

A real-valued symmetric matrix is orthogonally diagonalizable as , with diagonal and real valued and orthogonal. Verify that the following matrix is symmetric and then diagonalize it:

To diagonalize, first compute 's eigenvalues and place them in a diagonal matrix:

Next, compute the unit eigenvectors:

Then can be diagonalized with as previously, and :

A matrix is unitary if . Show that the matrix is unitary:

Confirm with UnitaryMatrixQ:

A real-valued matrix is called normal if . Normal matrices are the most general kind of matrix that can be unitarily diagonalized as with diagonal and unitary. All real symmetric matrices are normal because both sides of the equality are simply :

Show that the following matrix is normal and then diagonalize it:

Confirm using NormalMatrixQ:

A normal matrix like can be unitarily diagonalized using Eigensystem:

Unlike the case of a symmetric matrix, the diagonal matrix here is complex valued:

Normalizing the eigenvectors and putting them in columns gives a unitary matrix:

Confirm the diagonalization :

Show that real antisymmetric matrices and orthogonal matrices are normal and thus can be unitarily diagonalized. For orthogonal matrices, simply substitute in the definition to get the identity matrix on both sides: