orth

Calculate and verify the orthonormal basis vectors for the range of a full rank matrix.

Define a matrix and find the rank.

A = [1 0 1;-1 -2 0; 0 1 -1];
r = rank(A)

r = 
3

Because A is a square matrix of full rank, the orthonormal basis calculated by orth(A) matches the matrix U calculated in the singular value decomposition [U,S] = svd(A,"econ"). The reason is that the singular values of A are all nonzero.

Calculate the orthonormal basis for the range of A using orth.

Q = orth(A)

Q = 3×3

   -0.1200   -0.8097    0.5744
    0.9018    0.1531    0.4042
   -0.4153    0.5665    0.7118

The number of columns in Q is equal to rank(A). Because A is full rank, Q and A are the same size.

Verify that the basis, Q, is orthogonal and normalized within a reasonable error range.

E = norm(eye(r)-Q'*Q,"fro")

E = 
1.0857e-15

The error is on the order of eps.

Calculate and verify the orthonormal basis vectors for the range of a rank deficient matrix.

Define a singular matrix and find the rank.

A = [1 0 1; 0 1 0; 1 0 1];
r = rank(A)

r = 
2

Because A is rank deficient, the orthonormal basis calculated by orth(A) matches only the first r = 2 columns of matrix U calculated in the singular value decomposition [U,S] = svd(A,"econ"). The reason is that the singular values of A are not all nonzero.

Calculate the orthonormal basis for the range of A using orth.

Q = orth(A)

Q = 3×2

   -0.7071         0
         0    1.0000
   -0.7071         0

Because A is rank deficient, Q contains one fewer column than A.

When a matrix has small singular values, specify a tolerance to change which singular values are treated as zero.

Create a 7-by-7 Hilbert matrix. This matrix is full rank but has some small singular values.

H = hilb(7)

H = 7×7

    1.0000    0.5000    0.3333    0.2500    0.2000    0.1667    0.1429
    0.5000    0.3333    0.2500    0.2000    0.1667    0.1429    0.1250
    0.3333    0.2500    0.2000    0.1667    0.1429    0.1250    0.1111
    0.2500    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000
    0.2000    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909
    0.1667    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833
    0.1429    0.1250    0.1111    0.1000    0.0909    0.0833    0.0769

Calculate an orthonormal basis for the range of H. Because H is full rank, Q and H are the same size.

Q = orth(H)

Q = 7×7

   -0.7332    0.6232    0.2608   -0.0752    0.0160   -0.0025    0.0002
   -0.4364   -0.1631   -0.6706    0.5268   -0.2279    0.0618   -0.0098
   -0.3198   -0.3215   -0.2953   -0.4257    0.6288   -0.3487    0.0952
   -0.2549   -0.3574    0.0230   -0.4617   -0.2004    0.6447   -0.3713
   -0.2128   -0.3571    0.2337   -0.1712   -0.4970   -0.1744    0.6825
   -0.1831   -0.3446    0.3679    0.1827   -0.1849   -0.5436   -0.5910
   -0.1609   -0.3281    0.4523    0.5098    0.4808    0.3647    0.1944

Now, calculate the orthonormal basis vectors again, but specify a tolerance of 1e-4. This tolerance leads to orth treating three of the singular values as zeros, so the orthonormal basis has only four columns.

Qtol = orth(H,1e-4)

Qtol = 7×4

   -0.7332    0.6232    0.2608   -0.0752
   -0.4364   -0.1631   -0.6706    0.5268
   -0.3198   -0.3215   -0.2953   -0.4257
   -0.2549   -0.3574    0.0230   -0.4617
   -0.2128   -0.3571    0.2337   -0.1712
   -0.1831   -0.3446    0.3679    0.1827
   -0.1609   -0.3281    0.4523    0.5098