Compute Orthonormal Basis

Compute an orthonormal basis of the range of this matrix. Because these numbers are not symbolic objects, you get floating-point results.

A = [2 -3 -1; 1 1 -1; 0 1 -1];
B = orth(A)

B =
   -0.9859   -0.1195    0.1168
    0.0290   -0.8108   -0.5846
    0.1646   -0.5729    0.8029

Now, convert this matrix to a symbolic object, and compute an orthonormal basis:

A = sym([2 -3 -1; 1 1 -1; 0 1 -1]);
B = orth(A)

B =
[ (2*5^(1/2))/5, -6^(1/2)/6, -(2^(1/2)*15^(1/2))/30]
[     5^(1/2)/5,  6^(1/2)/3,  (2^(1/2)*15^(1/2))/15]
[             0,  6^(1/2)/6,  -(2^(1/2)*15^(1/2))/6]

You can use double to convert this result to the double-precision numeric form. The resulting matrix differs from the matrix returned by the MATLAB^® orth function because these functions use different versions of the Gram-Schmidt orthogonalization algorithm:

double(B)

ans =
    0.8944   -0.4082   -0.1826
    0.4472    0.8165    0.3651
         0    0.4082   -0.9129

Verify that B'*B = I, where I is the identity matrix:

B'*B

ans =
[ 1, 0, 0]
[ 0, 1, 0]
[ 0, 0, 1]

Now, verify that the 2-norm of each column of B is 1:

norm(B(:, 1))
norm(B(:, 2))
norm(B(:, 3))

ans =
1
 
ans =
1
 
ans =
1

Compute Real Orthonormal Basis

Compute an orthonormal basis of this matrix using 'real' to avoid complex conjugates:

syms a
A = [a 1; 1 a];
B = orth(A,'real')

B =
[ a/(a^2 + 1)^(1/2),    -(a^2 - 1)/((a^2 + 1)*((a^2 -...
 1)^2/(a^2 + 1)^2 + (a^2*(a^2 - 1)^2)/(a^2 + 1)^2)^(1/2))]
[ 1/(a^2 + 1)^(1/2), (a*(a^2 - 1))/((a^2 + 1)*((a^2 -...
 1)^2/(a^2 + 1)^2 + (a^2*(a^2 - 1)^2)/(a^2 + 1)^2)^(1/2))]

Compute Orthogonal Basis by Skipping Normalization

Compute an orthogonal basis of this matrix using 'skipnormalization'. The lengths of the resulting vectors (the columns of matrix B) are not required to be 1

syms a
A = [a 1; 1 a];
B = orth(A,'skipnormalization')

B =
[ a,               -(a^2 - 1)/(a*conj(a) + 1)]
[ 1, -(conj(a) - a^2*conj(a))/(a*conj(a) + 1)]

Compute Real Orthogonal Basis

Compute an orthogonal basis of this matrix using 'skipnormalization' and 'real':

syms a
A = [a 1; 1 a];
B = orth(A,'skipnormalization','real')

B =
[ a,    -(a^2 - 1)/(a^2 + 1)]
[ 1, (a*(a^2 - 1))/(a^2 + 1)]