Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: [A[:i,:i].determinant() for i in range(1,A.nrows()+1)]
[3, -3, -15, 30, -30]
sage: u = vector(QQ, [2, 2, 0, 1, 0])
sage: u.row()*A*u
(-3)
A real symmetric matrix with a singular leading principal submatrix, that is therefore not positive definite.
The vector u makes the quadratic form zero.
sage: A = matrix(QQ, [[21, 15, 12, -2],
... [15, 12, 9, 6],
... [12, 9, 7, 3],
... [-2, 6, 3, 8]])
sage: A.is_positive_definite()
False
sage: [A[:i,:i].determinant() for i in range(1,A.nrows()+1)]
[21, 27, 0, -75]
sage: u = vector(QQ, [1,1,-3,0])
sage: u.row()*A*u
(0)
An Hermitian matrix that is positive definite.
sage: C. = NumberField(x^2 + 1)
sage: A = matrix(C, [[ 23, 17*I + 3, 24*I + 25, 21*I],
... [ -17*I + 3, 38, -69*I + 89, 7*I + 15],
... [-24*I + 25, 69*I + 89, 976, 24*I + 6],
... [ -21*I, -7*I + 15, -24*I + 6, 28]])
sage: A.is_positive_definite()
True
sage: _, d = A.indefinite_factorization(algorithm=’hermitian’)
sage: d
(23, 576/23, 89885/144, 142130/17977)
sage: [A[:i,:i].determinant() for i in range(1,A.nrows()+1)]
[23, 576, 359540, 2842600]
An Hermitian matrix that is not positive definite. The vector u makes the quadratic form negative.
sage: C. = QuadraticField(-1)
sage: B = matrix(C, [[ 2, 4 - 2*I, 2 + 2*I],
... [4 + 2*I, 8, 10*I],
... [2 - 2*I, -10*I, -3]])
sage: B.is_positive_definite()
False
sage: _, d = B.indefinite_factorization(algorithm=’hermitian’)
sage: d
(2, -2, 3)
sage: [B[:i,:i].determinant() for i in range(1,B.nrows()+1)]
[2, -4, -12]
sage: u = vector(C, [-5 + 10*I, 4 - 3*I, 0])
sage: u.row().conjugate()*B*u
(-50)
A positive definite matrix over an algebraically closed field.
sage: A = matrix(QQbar, [[ 2, 4 + 2*I, 6 - 4*I],
... [ -2*I + 4, 11, 10 - 12*I],
... [ 4*I + 6, 10 + 12*I, 37]])
sage: A.is_positive_definite()
190 Chapter 7. Base class for matrices, part 2