How can we generate an N-dimensional unitary column vector?

I would like to generate an N-dimesnional Unitary Column Vector using MATLAB.

[Jos (10584)]

What is your mathematical definition of a N-dimensional unitary matrix?

Note that the conjugate transpose is onyl properly defined for 2D matrices ...

[To So]

N-Dimensional Unitary Column Vector. Sorry about the wrong question title.

[Walter Roberson]

What is an N-Dimensional Unitary Column Vector ?

[To So]

I am trying to implement 2D PCA. This is the first step.

[Walter Roberson]

I am not certain, but I think it is indicating a column vector with n components whose euclidean length is 1. If you generate any column vector with n components (that is not all zero) and divide by norm() of the vector, the result would have that property.

[To So]

clc;

X = (rand(4,1)+ i*rand(4,1));

Xn = norm(X);

Xf = X/Xn;

Xf1 = conj(Xf);

U = Xf*Xf1';

I tried this code, but I'm not getting the desired result

[Walter Roberson]

Xf1 = Xf';
U = Xf1*Xf;

gives 1, so the conjugate transpose of Xf left multiplied by Xf gives a unitary matrix.

If you try for

  x = [x1; x2; x3; x4]
 A = x * x'

hoping A == eye(4), then

 A = [x1 * conj(x1), x1 * conj(x2), x1 * conj(x3), x1 * conj(x4)
      x2 * conj(x1), x2 * conj(x2), x2 * conj(x3), x2 * conj(x4)
      x3 * conj(x1), x3 * conj(x2), x3 * conj(x3), x3 * conj(x4)
      x4 * conj(x1), x4 * conj(x2), x4 * conj(x3), x4 * conj(x4)]
 

The top left must = 1, so conj(x1) = 1/x1 . Substitute that in,

 A = [x1 * 1/x1, x1 * conj(x2), x1 * conj(x3), x1 * conj(x4)
      x2 * 1/x1, x2 * conj(x2), x2 * conj(x3), x2 * conj(x4)
      x3 * 1/x1, x3 * conj(x2), x3 * conj(x3), x3 * conj(x4)
      x4 * 1/x1, x4 * conj(x2), x4 * conj(x3), x4 * conj(x4)]
  

Now the remaining entries in the first column must be 0, so x2/x1 == 0, x3/x1 == 0, x4/x1 == 0. That can only be true if x1 is infinite (in which case we made an invalid substitution), or if x2, x3, x4 are all 0. So make x2, x3, x4 all 0. But then look in the bottom right corner, x4 * conj(x4). With x4 0, conj(x4) is 0, and x4 * conj(x4) = 0, rather than the 1 required. And if x1 was infinite then x1 * conj(x1) would be infinite rather than 1, so we rule out the x1 infinite case.

Therefore it is not possible to construct a column vector of length 4 such that a right multiplication of it by its complex conjugate gives eye(4).

The left multiplication giving 1 is the best we can do.

[John D'Errico]

As far as implementing 2-d PCA on your own, that is a BAD idea.

If you have no idea how to use MATLAB sufficiently to be able to do the computations I gave in my answer, then you need to use code written by a professional. So use tools like PCA from the stats toolbox.

Answer 1

[John D'Errico]

Confusing. Usually that means you are also confused about what you really want.

x = rand(4,1) +i*rand(4,1);
x = x/norm(x)
x = x/norm(x)
x =
           0.56057676215826 +    0.0425294484210577i
          0.292261237849278 +     0.563194470925906i
          0.351888409287552 +      0.18104942066974i
          0.310227988205874 +     0.168803277611609i
>> x'*x
ans =
                         1

So x is a vector of length 4, having norm 1, so unitary. It is not a matrix. Are you looking to create a matrix? Must it be complex? Are you looking to create a random matrix? As I said, your question is highly confusing.

format short g
>> x = rand(4,4) +i*rand(4,4);
>> x = orth(x)
x =
     -0.10738 -    0.23467i      0.35584 +   0.075661i     -0.29957 +   0.017908i     0.074794 -    0.83988i
     -0.35853 -    0.54037i     -0.38986 +    0.24156i     -0.37517 +    0.43408i     0.022772 +    0.19852i
     -0.30096 -    0.34939i     0.075507 +    0.40693i      0.23797 -    0.65141i     -0.36004 +   0.073921i
     -0.26064 -    0.48194i      0.32675 -    0.61583i      0.31589 +  0.0032239i      0.29479 +     0.1646i

So a 4x4 random matrix, such that the columns are orthogonal, unitary, and complex. Actually, the rows also have that same property. So both x'*x and x*x' will both yield an identity matrix.

x'*x
ans =
            1 +          0i  -1.6653e-16 - 1.1102e-16i  -1.3878e-17 +          0i  -2.7756e-17 + 9.7145e-17i
  -1.6653e-16 + 1.1102e-16i            1 +          0i   1.3878e-16 + 8.3267e-17i    2.047e-16 + 5.5511e-17i
  -1.3878e-17 +          0i   1.3878e-16 - 8.3267e-17i            1 +          0i   1.1102e-16 - 2.0817e-17i
  -2.7756e-17 - 9.7145e-17i    2.047e-16 - 5.5511e-17i   1.1102e-16 + 2.0817e-17i            1 +          0i

x*x'
ans =
            1 +          0i   8.3267e-17 - 2.7756e-17i  -1.3878e-17 + 1.1102e-16i   2.7756e-17 - 3.8858e-16i
   8.3267e-17 + 2.7756e-17i            1 +          0i   2.6021e-17 - 1.3878e-16i            0 + 3.4694e-16i
  -1.3878e-17 - 1.1102e-16i   2.6021e-17 + 1.3878e-16i            1 +          0i   1.5266e-16 - 1.3878e-16i
   2.7756e-17 + 3.8858e-16i            0 - 3.4694e-16i   1.5266e-16 + 1.3878e-16i            1 +          0i

Note that the off-diagonal terms are zero to within floating point trash for a double.