Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: A = matrix(RDF, [[1, -1],
... [1, 1]])
sage: A.is_normal()
True
sage: not A.is_hermitian() and not A.is_skew_symmetric()
True
Sage has several fields besides the entire complex numbers where conjugation is non-trivial.
sage: F. = QuadraticField(-7)
sage: C = matrix(F, [[-2*b - 3, 7*b - 6, -b + 3],
... [-2*b - 3, -3*b + 2, -2*b],
... [ b + 1, 0, -2]])
sage: C = C*C.conjugate_transpose()
sage: C.is_normal()
True
A square, empty matrix is trivially normal.
sage: A = matrix(CDF, 0, 0)
sage: A.is_normal()
True
Rectangular matrices are never normal, no matter which algorithm is requested.
sage: A = matrix(CDF, 3, 4)
sage: A.is_normal()
False
TESTS:
The tolerance must be strictly positive.
sage: A = matrix(RDF, 2, range(4))
sage: A.is_normal(tol = -3.1)
Traceback (most recent call last):
...
ValueError: tolerance must be positive, not -3.1
The algorithm keyword gets checked.
sage: A = matrix(RDF, 2, range(4))
sage: A.is_normal(algorithm=’junk’)
Traceback (most recent call last):
...
ValueError: algorithm must be ’naive’ or ’orthonormal’, not junk
AUTHOR:
•Rob Beezer (2011-03-31)
is_positive_definite()
Determines if a matrix is positive definite.
A matrix A is positive definite if it is square, is Hermitian (which reduces to symmetric in the real case),
and for every nonzero vector ⃗x,
⃗x ∗ A⃗x > 0
where ⃗x ∗ is the conjugate-transpose in the complex case and just the transpose in the real case. Equivalently,
a positive definite matrix has only positive eigenvalues and only positive determinants of leading
principal submatrices.
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