Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•dimensions - the list of dimensions corresponding to each eigenspace (default=None).
OUTPUT:
A square, diagonalizable, matrix with only integer entries. The eigenspaces of this matrix, if computed by hand,
give basis vectors with only integer entries.
Note: It is easiest to use this function via a call to the random_matrix() function with the
algorithm=’diagonalizable’ keyword. We provide one example accessing this function directly,
while the remainder will use this more general function.
EXAMPLES:
A diagonalizable matrix, size 5.
sage: from sage.matrix.constructor import random_diagonalizable_matrix
sage: matrix_space = sage.matrix.matrix_space.MatrixSpace(QQ, 5)
sage: A=random_diagonalizable_matrix(matrix_space); A # random
[ 10 18 8 4 -18]
[ 20 10 8 4 -16]
[-60 -54 -22 -12 18]
[-60 -54 -24 -6 6]
[-20 -18 -8 -4 8]
sage: A.eigenvalues() # random
[10,6,2,-8,-10]
sage: S=A.right_eigenmatrix()[1]; S # random
[ 1 1 1 1 0]
[ 1 1 1 0 1]
[-3 -3 -4 -3 -3]
[-3 -4 -3 -3 -3]
[-1 -1 -1 -1 -1]
sage: S_inverse=S.inverse(); S_inverse # random
[ 1 1 1 1 -5]
[ 0 0 0 -1 3]
[ 0 0 -1 0 3]
[ 0 -1 0 0 -1]
[-1 0 0 0 -1]
sage: S_inverse*A*S # random
[ 10 0 0 0 0]
[ 0 6 0 0 0]
[ 0 0 2 0 0]
[ 0 0 0 -8 0]
[ 0 0 0 0 -10]
A diagonalizable matrix with eigenvalues and dimensions designated, with a check that if eigenvectors were
calculated by hand entries would all be integers.
sage: B=random_matrix(QQ, 6, algorithm=’diagonalizable’, eigenvalues=[-12,4,6],dimensions=[2,3,1
[ -52 32 240 -464 -96 -520]
[ 6 4 -48 72 36 90]
[ 46 -32 -108 296 -12 274]
[ 24 -16 -64 164 0 152]
[ 18 -16 0 72 -48 30]
[ 2 0 -16 24 12 34]
sage: all([x in ZZ for x in (B-(-12*identity_matrix(6))).rref().list()])
True
sage: all([x in ZZ for x in (B-(4*identity_matrix(6))).rref().list()])
True
sage: all([x in ZZ for x in (B-(6*identity_matrix(6))).rref().list()])
42 Chapter 2. Matrix Constructor