linalg::charmat
--
characteristic matrixlinalg::charmat
(A, x)
returns the
characteristic matrix x*I - A of the n x n matrix
A, where I denotes the n x n identity
matrix.
linalg::charmat(A, x)
A |
- | a square matrix of a domain of category Cat::Matrix |
x |
- | an indeterminate |
a matrix of the domain
Dom::Matrix(Dom::DistributedPolynomial([x],R))
, where
R
is the component ring of A
.
A
must be a commutative ring,
i.e., a domain of category Cat::CommutativeRing
.evalp(M, x =
u)
. See example 2.We define a matrix over the rational numbers:
>> A := Dom::Matrix(Dom::Rational)([[1, 2], [3, 4]])
+- -+ | 1, 2 | | | | 3, 4 | +- -+
and compute the characteristic matrix of A in the variable x:
>> MA := linalg::charmat(A, x)
+- -+ | x - 1, -2 | | | | -3, x - 4 | +- -+
The determinant of the matrix MA
is a
polynomial in x, the characteristic polynomial of the matrix
A:
>> pA := linalg::det(MA)
2 x - 5 x - 2
>> domtype(pA)
Dom::DistributedPolynomial([x], Dom::Rational, LexOrder)
Of course, we can compute the characteristic polynomial
of A directly via linalg::charpoly
:
>> linalg::charpoly(A, x)
2 x - 5 x - 2
The result is of the same domain type as the polynomial
pA
.
We define a matrix over the complex numbers:
>> B := Dom::Matrix(Dom::Complex)([[1 + I, 1], [1, 1 - I]])
+- -+ | 1 + I, 1 | | | | 1, 1 - I | +- -+
The characteristic matrix of B
in the
variable z is:
>> MB := linalg::charmat(B, z)
+- -+ | z - (1 + I), -1 | | | | -1, z - (1 - I) | +- -+
We evaluate MB
at z=I and get
the matrix:
>> evalp(MB, z = I)
+- -+ | -1, -1 | | | | -1, - 1 + 2 I | +- -+
Note that this is a matrix of the domain type
Dom::Matrix(Dom::Complex)
:
>> domtype(%)
Dom::Matrix(Dom::Complex)
linalg::charMatrix