orth

Orthonormal basis for range of matrix

Syntax

Q = orth(A)

Q = orth(A,tol)

Description

Q = orth(A) returns an orthonormal basis for the range of A. The columns of matrix Q are vectors that span the range of A. The number of columns in Q is equal to the rank of A.

Q = orth(A,tol) also specifies a tolerance. Singular values of A less than or equal to tol are treated as zero, which can affect the number of columns in Q.

Examples

Basis for Full Rank Matrix

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.

Basis for Rank Deficient Matrix

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.

Specify Tolerance for Basis Vectors

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

Input Arguments

A — Input matrix

matrix

Input matrix.

Data Types: single | double
Complex Number Support: Yes

tol — Singular value tolerance

scalar

Singular value tolerance, specified as a real numeric scalar. Singular values of A less than or equal to the tolerance are treated as zero, which affects the number of column space vectors returned by orth. The default tolerance is max(size(A)) * eps(norm(A)).

More About

Range

The column space, or range, of a matrix A is the collection of all linear combinations of the columns of A. Any vector, b, that is a solution to the linear equation A*x = b is included in the range of A because you can also write it as a linear combination of the columns of A.

Rank

The rank of a matrix is equal to the dimension of the range, and is equal to the number of nonzero singular values.

Algorithms

The orthonormal basis for the range of A is obtained from U in the singular value decomposition [U,S] = svd(A,"econ"). If r = rank(A,tol), then the first r columns of U form an orthonormal basis for the range of A.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Generated code can return a different basis than MATLAB^® returns.

• Code generation does not support sparse matrix inputs for this function.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Refer to the usage notes and limitations in the C/C++ Code Generation section. The same usage notes and limitations apply to GPU code generation.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

The orth function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The orth function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

The orth function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2022a: Specify tolerance

Use the tol argument to specify a tolerance threshold for the singular values used to form the orthonormal basis for the range of A. Singular values of the input matrix less than or equal to the tolerance are treated as zero.