Sage Reference Manual: Matrices and Spaces of Matrices

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2•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 thealgorithm=’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_matrixsage: 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 werecalculated 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()])Truesage: all([x in ZZ for x in (B-(4*identity_matrix(6))).rref().list()])Truesage: all([x in ZZ for x in (B-(6*identity_matrix(6))).rref().list()])42 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2Truesage: S=B.right_eigenmatrix()[1]; S_inverse=S.inverse(); S_inverse*B*S # random[ 6 0 0 0 0 0][ 0 -12 0 0 0 0][ 0 0 -12 0 0 0][ 0 0 0 4 0 0][ 0 0 0 0 4 0][ 0 0 0 0 0 4]TESTS:Eigenvalues must all be integers.sage: random_matrix(QQ,3,algorithm=’diagonalizable’, eigenvalues=[2+I,2-I,2],dimensions=[1,1,1])Traceback (most recent call last):...TypeError: eigenvalues must be integers.Diagonal matrices must be square.sage: random_matrix(QQ, 5, 7, algorithm=’diagonalizable’, eigenvalues=[-5,2,-3], dimensions=[1,1Traceback (most recent call last):...TypeError: a diagonalizable matrix must be square.A list of eigenvalues must be accompanied with a list of dimensions.sage: random_matrix(QQ,10,algorithm=’diagonalizable’,eigenvalues=[4,8])Traceback (most recent call last):...ValueError: the list of eigenvalues must have a list of dimensions corresponding to each eigenvaA list of dimensions must be accompanied with a list of eigenvalues.sage: random_matrix(QQ, 10,algorithm=’diagonalizable’,dimensions=[2,2,4,2])Traceback (most recent call last):...ValueError: the list of dimensions must have a list of corresponding eigenvalues.The sum of the eigenvalue dimensions must equal the size of the matrix.sage: random_matrix(QQ,12,algorithm=’diagonalizable’,eigenvalues=[4,2,6,-1],dimensions=[2,3,5,1]Traceback (most recent call last):...ValueError: the size of the matrix must equal the sum of the dimensions.Each eigenspace dimension must be at least 1.sage: random_matrix(QQ,9,algorithm=’diagonalizable’,eigenvalues=[-15,22,8,-4,90,12],dimensions=[Traceback (most recent call last):...ValueError: eigenspaces must have a dimension of at least 1.Each eigenvalue must have a corresponding eigenspace dimension.sage: random_matrix(QQ,12,algorithm=’diagonalizable’,eigenvalues=[4,2,6,-1],dimensions=[4,3,5])Traceback (most recent call last):...ValueError: each eigenvalue must have a corresponding dimension and each dimension a correspondiEach dimension must have an eigenvalue paired to it.43
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Sage Reference Manual: Matrices and Spaces of Matrices

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