Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 SubspaceAngle

Source Notebook

Compute the angle between two subspaces

Contributed by: Jan Mangaldan

ResourceFunction["SubspaceAngle"][u,v]

gives the angle between two subspaces specified by the columns of the matrices u and v.

Details and Options

The matrices u and v must be numerical matrices with the same number of rows.
Vectors are treated as column matrices.
If the elements of u or v are exact numbers, ResourceFunction["SubspaceAngle"] begins by applying N to them.
ResourceFunction["SubspaceAngle"] gives an angle in radians.

Examples

Basic Examples (1) 

Compute the angle between two subspaces:

In[1]:=
a = {{1., 1.}, {1., 2.}, {1., 3.}}; b = {{7., 3.}, {1., 1.}, {5., -8.}}; ResourceFunction["SubspaceAngle"][a, b]
Out[3]=

Scope (2) 

Find the subspace angle between a vector and a matrix:

In[4]:=
ResourceFunction["SubspaceAngle"][{1., 2., 3.}, ( { {1., 2.}, {1., 3.}, {1., 5.} } )]
Out[4]=

Find the subspace angle between different sets of columns of a complex matrix:

In[5]:=
fm = FourierMatrix[8, FourierParameters -> {1, -1}, WorkingPrecision -> 20];
In[6]:=
ResourceFunction["SubspaceAngle"][fm[[All, 1 ;; 3]], fm[[All, 5 ;; 8]]]
Out[6]=

Properties and Relations (2) 

For two vectors, SubspaceAngle gives the same result as VectorAngle:

In[7]:=
ResourceFunction["SubspaceAngle"][{1., 2., 3.}, {1., 1., 1.}]
Out[7]=
In[8]:=
VectorAngle[{1., 2., 3.}, {1., 1., 1.}]
Out[8]=

Use SubspaceAngle to compute the angle between two planes by specifying the vectors spanning each plane as matrix columns:

In[9]:=
ResourceFunction["SubspaceAngle"][( { {0., 1.}, {1., 0.}, {0., 1.} } ), ( { {0., 1.}, {1., 0.}, {0., 0.} } )]
Out[9]=

Use DihedralAngle to compute the same result:

In[10]:=
DihedralAngle[{{0., 0., 0.}, {0., 1., 0.}}, {{1., 0., 1.}, {1., 0., 0.}}]
Out[10]=

Version History

  • 1.0.0 – 08 February 2023

Source Metadata

  • Citation:
    • Björck Ȧ., Golub G.H., "Numerical Methods for Computing Angles between Linear Subspaces." Mathematics of Computation, vol. 27, no. 123, 579-594, 1973.
      DOI: 10.1090/S0025-5718-1973-0348991-3
    • Knyazev A.V., Argentati M.E., "Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates." SIAM Journal on Scientific Computing, vol. 23, no. 6, 2008-2040, 2002.
      DOI: 10.1137/s1064827500377332

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