# Theorems (or conjectures) for the theory of proveit.linear_algebra.matrices¶

In [1]:
import proveit
# Prepare this notebook for defining the theorems of a theory:
%theorems_notebook # Keep this at the top following 'import proveit'.

from proveit import (a, b, b_of_j, b_of_k, c, d, f, g, i, j, k, m, n, t,
x, y, z, fj, gj, fy, A, P, U, V, W, S, alpha, beta)
from proveit import Function, ExprRange, IndexedVar
from proveit.core_expr_types import (bj, a_1_to_m, x_1_to_i, x_1_to_m,
y_1_to_j, z_1_to_k, z_1_to_n)
from proveit.core_expr_types import (
a_1_to_i, b_1_to_j, c_1_to_j, c_1_to_k, d_1_to_k)
from proveit.logic import Equals, Forall, Implies, Iff, InSet, InClass, SubsetEq
from proveit.numbers import Add, Mult, Exp, Sum
from proveit.numbers import one
from proveit.numbers import (Integer, Interval, Natural, NaturalPos,
Rational, Real, Complex)
from proveit.linear_algebra import (
VecSpaces, LinMap, ScalarMult, MatrixSpace,
TensorProd, MatrixMult, Unitary, SpecialUnitary)

In [2]:
%begin theorems

Defining theorems for theory 'proveit.linear_algebra.matrices'
Subsequent end-of-cell assignments will define theorems
'%end theorems' will finalize the definitions


### Matrix spaces on number fields are vector spaces¶

In [3]:
rational_matrix_space_is_vec_space = Forall(
(m, n), InClass(MatrixSpace(Rational, m, n),
VecSpaces(Rational)),
domain=NaturalPos)

rational_matrix_space_is_vec_space (conjecture without proof):

In [4]:
real_matrix_space_is_vec_space = Forall(
(m, n), InClass(MatrixSpace(Real, m, n),
VecSpaces(Real)),
domain=NaturalPos)

real_matrix_space_is_vec_space (conjecture without proof):

In [5]:
complex_matrix_space_is_vec_space = Forall(
(m, n), InClass(MatrixSpace(Complex, m, n),
VecSpaces(Complex)),
domain=NaturalPos)

complex_matrix_space_is_vec_space (conjecture without proof):

In [6]:
eigen_pow = (
Forall(k,
Forall(b,
Forall((A, x),
Implies(Equals(MatrixMult(A, x), ScalarMult(b, x)),
Equals(MatrixMult(Exp(A, k), x), ScalarMult(Exp(b, k), x)))),
domain=Complex), domain=NaturalPos))

eigen_pow (conjecture without proof):

In [7]:
unitaries_are_matrices = Forall(
n, SubsetEq(Unitary(n), MatrixSpace(Complex, n, n)),
domain=NaturalPos)

unitaries_are_matrices (conjecture without proof):

In [8]:
special_unitaries_are_matrices = Forall(
n, SubsetEq(SpecialUnitary(n), MatrixSpace(Complex, n, n)),
domain=NaturalPos)

special_unitaries_are_matrices (conjecture without proof):

In [9]:
%end theorems

These theorems may now be imported from the theory package: proveit.linear_algebra.matrices