dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day! A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

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^* $ (asterisk) 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.

dCode retains ownership of the online 'Conjugate Transpose Matrix' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script, copy-paste, or API access will be for free, same for Conjugate Transpose Matrix download for offline use on PC, tablet, iPhone or Android !

Need Help ?

Please, check our community Discord for help requests!