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Conjugate Transpose Matrix

Tool to calculate adjoint matrix (or Hermitian transpose). The adjoint matrix is the transpose of the conjugate matrix of a matrix M.

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Conjugate Transpose Matrix -

Tag(s) : Matrix

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Conjugate Transpose Matrix

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Conjugate Transpose Matrix Calculator

Tool to calculate adjoint matrix (or Hermitian transpose). The adjoint matrix is the transpose of the conjugate matrix of a matrix M.

Answers to Questions

What is a matrix conjugate transpose? (Definition)

The conjugate transpose matrix is the name given to the transpose of the conjugate of a complex matrix (or of the conjugate of the transposed matrix), it is denoted \( M^* \) or, more rare notation, with a dagger † \( M^\dagger \).

How to calculate the conjugate transpose of a matrix? (Formula)

\( M=[a_{ij}] \) is a matrix with complex elements, the conjugate transpose matrix is computed with the formula $$ M^* = \overline{M}^T = \overline{M^T} = [\overline{a_{ij}}]^T $$

Example: The conjugate transpose 2x2 matrix \( M^* \) of the matrix \( M \) is calculated: $$ M=\begin{pmatrix} 2 & 1-i & 0 \\ 1 & 2+i & -i \end{pmatrix} \Rightarrow M^*= \begin{pmatrix} 2 & 1 \\ 1+i & 2-i \\ 0 & i \end{pmatrix} $$

On dCode, use the character i to represent the imaginary unit \( i \) of complex numbers.

What is the hermitian transpose?

Hermitian transpose is another name of the conjugate transpose matrix, mainly used on linear function spaces. Other names used : Hermitian conjugate, bedaggered matrix or transjugate.

What is the adjoint matrix?

In English, the conjugate transposed matrix is sometimes called adjoint matrix but it is not the same as the generally defined adjoint matrix.

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