How to resolve the algorithm Conjugate transpose step by step in the J programming language

Published on 12 May 2024 09:40 PM
#J

How to resolve the algorithm Conjugate transpose step by step in the J programming language

Table of Contents

Problem Statement

Suppose that a matrix

M

{\displaystyle M}

contains complex numbers. Then the conjugate transpose of

M

{\displaystyle M}

is a matrix

M

H

{\displaystyle M^{H}}

containing the complex conjugates of the matrix transposition of

M

{\displaystyle M}

.

This means that row

j

{\displaystyle j}

, column

i

{\displaystyle i}

of the conjugate transpose equals the complex conjugate of row

i

{\displaystyle i}

, column

j

{\displaystyle j}

of the original matrix.

In the next list,

M

{\displaystyle M}

must also be a square matrix.

Given some matrix of complex numbers, find its conjugate transpose. Also determine if the matrix is a:

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Conjugate transpose step by step in the J programming language

Source code in the j programming language

   ct =: +@|:                      NB.  Conjugate transpose (ct A is A_ct)


   X         =: +/ . *             NB. Matrix Multiply (x)

   HERMITIAN =:  3 2j1 ,: 2j_1 1  
   (-: ct) HERMITIAN               NB.  A_ct = A
1

   NORMAL    =:  1 1 0 , 0 1 1 ,: 1 0 1
   ((X~ -: X) ct) NORMAL           NB. A_ct x A = A x A_ct
1

   UNITARY   =:  (-:%:2) * 1 1 0 , 0j_1 0j1 0 ,: 0 0 0j1 * %:2
   (ct -: %.)  UNITARY             NB.  A_ct = A^-1
1


   HERMITIAN;NORMAL;UNITARY
+--------+-----+--------------------------+
|   3 2j1|1 1 0|   0.707107   0.707107   0|
|2j_1   1|0 1 1|0j_0.707107 0j0.707107   0|
|        |1 0 1|          0          0 0j1|
+--------+-----+--------------------------+
   NB. In J, PjQ is P + Q*i and the 0.7071... is sqrt(2)

   hermitian=: -: ct
   normal =: (X~ -: X) ct
   unitary=: ct -: %.

   (hermitian,normal,unitary)&.>HERMITIAN;NORMAL;UNITARY
+-----+-----+-----+
|1 1 0|0 1 0|0 1 1|
+-----+-----+-----+

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