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Conjugate Transpose Matrix

Tool to calculate the conjugate transpose matrix (or Hermitian transpose matrix), the transpose of the conjugate matrix of a complex matrix M.

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Conjugate Transpose Matrix -

Tag(s) : Matrix

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Conjugate Transpose Matrix

Conjugate Transpose Matrix Calculator

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Hermitian Matrix Checker

Calculate the Conjugate Transpose matrix if it is equal to the initial matrix then the matrix is Hermitian.

Answers to Questions (FAQ)

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^* $ (asterisk) or, more rare notation, with a dagger † $ M^\dagger $.

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

Taking $ M=[a_{ij}] $ 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: $$ M=\begin{bmatrix} 2 & 1-i & 0 \\ 1 & 2+i & -i \end{bmatrix} \Rightarrow M^*= \begin{bmatrix} 2 & 1 \\ 1+i & 2-i \\ 0 & i \end{bmatrix} $$

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 erroneously called adjoint matrix but it is not the same matrix.

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