Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
The QR decoposition will produce a unitary matrix as Q and the SVD decomposition will create two
unitary matrices, U and V.
sage: A = matrix(CDF, [[ 1 - I, -3*I, -2 + I, 1, -2 + 3*I],
... [ 1 - I, -2 + I, 1 + 4*I, 0, 2 + I],
... [ -1, -5 + I, -2 + I, 1 + I, -5 - 4*I],
... [-2 + 4*I, 2 - I, 8 - 4*I, 1 - 8*I, 3 - 2*I]])
sage: Q, R = A.QR()
sage: Q.is_unitary()
True
sage: U, S, V = A.SVD()
sage: U.is_unitary(algorithm=’naive’)
True
sage: U.is_unitary(algorithm=’orthonormal’)
True
sage: V.is_unitary(algorithm=’naive’) # not tested - known bug (trac #11248)
True
If we make the tolerance too strict we can get misleading results.
sage: A = matrix(RDF, 10, 10, [1/(i+j+1) for i in range(10) for j in range(10)])
sage: Q, R = A.QR()
sage: Q.is_unitary(algorithm=’naive’, tol=1e-16)
False
sage: Q.is_unitary(algorithm=’orthonormal’, tol=1e-17)
False
Rectangular matrices are not unitary/orthogonal, even if their columns form an orthonormal set.
sage: A = matrix(CDF, [[1,0], [0,0], [0,1]])
sage: A.is_unitary()
False
The smallest cases. The Schur decomposition used by the orthonormal algorithm will fail on a matrix of
size zero.
sage: P = matrix(CDF, 0, 0)
sage: P.is_unitary(algorithm=’naive’)
True
sage: P = matrix(CDF, 1, 1, [1])
sage: P.is_unitary(algorithm=’orthonormal’)
True
sage: P = matrix(CDF, 0, 0,)
sage: P.is_unitary(algorithm=’orthonormal’)
Traceback (most recent call last):
...
ValueError: failed to create intent(cache|hide)|optional array-- must have defined dimension
TESTS:
sage: P = matrix(CDF, 2, 2)
sage: P.is_unitary(tol=’junk’)
Traceback (most recent call last):
...
TypeError: tolerance must be a real number, not junk
sage: P.is_unitary(tol=-0.3)
Traceback (most recent call last):
375