# conjugate transpose

Definition If $A$ is a complex matrix, then the conjugate transpose $A^{\ast}$ is the matrix $A^{\ast}=\bar{A}\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$, where $\bar{A}$ is the complex conjugate of $A$, and $A\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$ is the transpose of $A$.

It is clear that for real matrices, the conjugate transpose coincides with the transpose.

## 0.0.1 Properties

1. 1.

If $A$ and $B$ are complex matrices of same size, and $\alpha,\beta$ are complex constants, then

 $\displaystyle(\alpha A+\beta B)^{\ast}$ $\displaystyle=$ $\displaystyle\overline{\alpha}A^{\ast}+\overline{\beta}B^{\ast},$ $\displaystyle A^{\ast\ast}$ $\displaystyle=$ $\displaystyle A.$
2. 2.

If $A$ and $B$ are complex matrices such that $AB$ is defined, then

 $(AB)^{\ast}=B^{\ast}A^{\ast}.$
3. 3.

If $A$ is a complex square matrix , then

 $\displaystyle\det(A^{\ast})$ $\displaystyle=$ $\displaystyle\overline{\det{A}},$ $\displaystyle\operatorname{trace}(A^{\ast})$ $\displaystyle=$ $\displaystyle\overline{\operatorname{trace}{A}},$ $\displaystyle(A^{\ast})^{-1}$ $\displaystyle=$ $\displaystyle(A^{-1})^{\ast},$

where $\operatorname{trace}$ and $\operatorname{det}$ are the trace and the determinant operators, and ${}^{-1}$ is the inverse operator.

4. 4.

Suppose $\langle\cdot,\cdot\rangle$ is the standard inner product on $\mathbb{C}^{n}$. Then for an arbitrary complex $n\times n$ matrix $A$, and vectors $x,y\in\mathbb{C}^{n}$, we have

 $\langle Ax,y\rangle=\langle x,A^{\ast}y\rangle.$

## Notes

The conjugate transpose of $A$ is also called the adjoint matrix of $A$, the Hermitian conjugate of $A$ (whence one usually writes $A^{\ast}=A\hskip 1.13811pt^{\mbox{\scriptsize{H}}}\hskip 0.569055pt$). The notation $A^{\dagger}$ is also used for the conjugate transpose . In , $A^{\ast}$ is also called the tranjugate of $A$.

## References

• 1
• 2 M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965.
{"ft_lang_label":"__label__en","ft_lang_prob":0.79898274,"math_prob":0.9999938,"size":1516,"snap":"2021-04-2021-17","text_gpt3_token_len":402,"char_repetition_ratio":0.18849206,"word_repetition_ratio":0.008547009,"special_character_ratio":0.26912928,"punctuation_ratio":0.15957446,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000005,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-28T03:23:25Z\",\"WARC-Record-ID\":\"<urn:uuid:4673b1e5-b077-408a-815c-dc9386460377>\",\"Content-Length\":\"22989\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6716246d-087d-496e-9cef-f1687ef3f72d>\",\"WARC-Concurrent-To\":\"<urn:uuid:bba71618-322b-4378-bd8e-3ba3ba24bce1>\",\"WARC-IP-Address\":\"129.97.206.129\",\"WARC-Target-URI\":\"https://planetmath.org/ConjugateTranspose\",\"WARC-Payload-Digest\":\"sha1:4KD3Q6IIDWBLCTO7CPD4UMAC3WI3W57G\",\"WARC-Block-Digest\":\"sha1:ZEC7WBL62IGTJHKDELJXUPRHZUTDU4TN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610704835583.91_warc_CC-MAIN-20210128005448-20210128035448-00618.warc.gz\"}"}