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Hermitian transpose of a matrix

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htranspose(A) returns the Hermitian transpose AH of the matrix A (the complex conjugate of the transpose of A). The Hermitian transpose of the m×n matrix A is the n×m matrix B with .

If the input is a matrix of category Cat::Matrix, then internally linalg::htranspose computes the result. For arrays and hfarrays, htranspose uses other routines.

If the argument does not evaluate to a matrix of one of these types, the transpose is the conjugate of the input.


Example 1

Compute the transpose of the following real matrix. For real matrices, the Hermitian transpose coincides with the transpose:

A := matrix([[1, 2], [3, PI]])

transpose(A) = htranspose(A)

In general, this does not hold for complex matrices:

A := matrix([[1, I, 3 + I], [PI*I, 4, 5]])

transpose(A) <> htranspose(A)

delete A:

Example 2

Compute the product AHA of a matrix given by a hardware float array. This data type allows matrix multiplication using the operator *:

A := hfarray(1..2, 1..3, [[1, I, 3], [PI*I, 4, 5 + I]])

AH:= htranspose(A)

The product AHA is Hermitian:

AH*A = htranspose(AH*A)

delete A, AH:

Example 3

If the input does not evaluate to a matrix, then the transpose is the conjugate of the input:

htranspose(A) + 2*htranspose(B)



An object of the category Cat::Matrix, a two-dimensional array, or a two-dimensional hfarray.

Return Values

Object of the same domain type as A.

Overloaded By


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