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:

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:

is a random real matrix:

Find the QRDecomposition of :

is orthogonal, so its inverse is TemplateBox[{q}, Transpose]:

Reconstruct from the decomposition:

Compute the SchurDecomposition of a matrix :

The matrix is orthogonal, so its inverse is TemplateBox[{q}, Transpose]:

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 TemplateBox[{a}, Transpose].a:

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:

A symmetric matrix obeys s=TemplateBox[{s}, Transpose], an antisymmetric matrix a=-TemplateBox[{a}, Transpose]. This matrix is symmetric:

Confirm with SymmetricMatrixQ :

This matrix is antisymmetric:

Confirm with AntisymmetricMatrixQ :

A matrix is orthogonal if TemplateBox[{o}, Inverse]=TemplateBox[{o}, Transpose]. Check if the matrix is orthogonal:

Confirm that it is orthogonal using OrthogonalMatrixQ :

A real-valued symmetric matrix is orthogonally diagonalizable as s=o.d.TemplateBox[{o}, Transpose], 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 o=TemplateBox[{v}, Transpose]:

A matrix is unitary if TemplateBox[{u}, Inverse]=TemplateBox[{{(, TemplateBox[{u}, Transpose, SyntaxForm -> SuperscriptBox], )}}, Conjugate]. Show that the matrix is unitary:

Confirm with UnitaryMatrixQ :

A real-valued matrix is called normal if n.TemplateBox[{n}, Transpose]=TemplateBox[{n}, Transpose].n. Normal matrices are the most general kind of matrix that can be unitarily diagonalized as n=u.d.TemplateBox[{{(, TemplateBox[{u}, Transpose, SyntaxForm -> SuperscriptBox], )}}, Conjugate] 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 n=u.d.TemplateBox[{{(, TemplateBox[{u}, Transpose, SyntaxForm -> SuperscriptBox], )}}, Conjugate]:

Show that real antisymmetric matrices and orthogonal matrices are normal and thus can be unitarily diagonalized. For orthogonal matrices, simply substitute in the definition TemplateBox[{o}, Transpose]=TemplateBox[{o}, Inverse] to get the identity matrix on both sides:

For an antisymmetric matrix, both sides are simply :

Orthogonal matrices have eigenvalues that lie on the unit circle:

Antisymmetric matrices have pure imaginary eigenvalues:

Use Transpose to change data grouping in BarChart :

Use Transpose to swap the and axes in ListPlot3D :

This has the effect of reflecting the data across the line :

Multidimensionalize (in the tensor product sense) a one-dimensional list command:

For example, accumulate at all levels of an array:

Reverse at all levels of an array:

Import an RGB image:

Reverse the data at all levels, reflecting across the line and swapping red and blue channels: