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# Math 152: Intro to Mathematical Software
### 2017-02-03
### Kiran Kedlaya; University of California, San Diego
### adapted from lectures by William Stein, University of Washington

## **Lecture 11: Linear Algebra (part 2)**

Math 152: Intro to Mathematical Software

2017-02-03

Kiran Kedlaya; University of California, San Diego

adapted from lectures by William Stein, University of Washington

Lecture 11: Linear Algebra (part 2)

Announcements:

  • Last call for the week 4 feedback survey! (See Wednesday's handout for the link.)

  • If you are still on the waitlist and still want to enroll, see me ASAP.

Today: more examples from the sage linear algebra quickref guide.

u = vector(QQ, [5, 3/2, -1])  # dense vector
show(u)
(5, 32, −1)\displaystyle \left(5,\,\frac{3}{2},\,-1\right)
u[0]  # 0 based
5 
parent(u)
Vector space of dimension 3 over Rational Field 
# a *sparse* vector -- same functions, but underlying algorithms for sparse vectors (and matrices) use sparse data structures/algorithms
v = vector({2:4, 95:4, 1000000:3/4})
parent(v)
Sparse vector space of dimension 1000001 over Rational Field 
v[15]
v[1000000]
0 3/4 
w = vector(QQ, {3:5, 4:6, 1000000:0})
parent(w)
Sparse vector space of dimension 1000001 over Rational Field 
w1 = v+w
parent(w1)
Sparse vector space of dimension 1000001 over Rational Field 
w2 = vector({2:4, 95:4, 1000000:3/4, 3:5, 4:6})
print(w1 == w2)
True 
af5ae59-3cb3-49fe-bc7e-cb4e72cf6ec6︠



m = matrix(QQ, 10000, sparse=True)


m== 0
True 
m[0,100] = 4/5

m[0,99]
0 
m = matrix(QQ, 2, sparse=True)
m[0,1] = 3
print(m)
[0 3] [0 0] 
u = vector([1, 3/2, -1]); v = vector([1/3, 8, -2])
u.dot_product(v)
u.cross_product(v)
u.inner_product(v)
u.pairwise_product(v)
43/3 (5, 5/3, 15/2) 43/3 (1/3, 12, 2) 
u = identity_matrix(CC, 5); print(u)
[ 1.00000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 1.00000000000000 0.000000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 1.00000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 0.000000000000000 1.00000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 1.00000000000000] 
identity_matrix?

File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/matrix/special.py
Signature : identity_matrix(ring, n=0, sparse=False)
Docstring :
This function is available as identity_matrix(...) and
matrix.identity(...).

Return the n x n identity matrix over the given ring.

The default ring is the integers.

EXAMPLES:

   sage: M = identity_matrix(QQ, 2); M
   [1 0]
   [0 1]
   sage: M.parent()
   Full MatrixSpace of 2 by 2 dense matrices over Rational Field
   sage: M = identity_matrix(2); M
   [1 0]
   [0 1]
   sage: M.parent()
   Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
   sage: M.is_mutable()
   True
   sage: M = identity_matrix(3, sparse=True); M
   [1 0 0]
   [0 1 0]
   [0 0 1]
   sage: M.parent()
   Full MatrixSpace of 3 by 3 sparse matrices over Integer Ring
   sage: M.is_mutable()
   True

%octave
eye(5)
ans = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 
A = matrix(QQ, [[1,2],[3,4]])
A^3 - 2*A + 1
[ 36 50] [ 75 111] 
A = matrix(QQ, [[1,2],[3,4]])
f(x) = x^3 - 2*x + 1
f(A) # FAIL -- this should work, but doesn't
Error in lines 3-3 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 982, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/symbolic/expression.pyx", line 5066, in sage.symbolic.expression.Expression.__call__ (/projects/sage/sage-7.5/src/build/cythonized/sage/symbolic/expression.cpp:30530) return self._parent._call_element_(self, *args, **kwds) File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/symbolic/callable.py", line 464, in _call_element_ return SR(_the_element.substitute(**d)) File "sage/symbolic/expression.pyx", line 4972, in sage.symbolic.expression.Expression.substitute (/projects/sage/sage-7.5/src/build/cythonized/sage/symbolic/expression.cpp:29999) (<Expression>self.coerce_in(v))._gobj)) File "sage/symbolic/expression.pyx", line 2885, in sage.symbolic.expression.Expression.coerce_in (/projects/sage/sage-7.5/src/build/cythonized/sage/symbolic/expression.cpp:21670) return self._parent._coerce_(z) File "sage/structure/parent_old.pyx", line 240, in sage.structure.parent_old.Parent._coerce_ (/projects/sage/sage-7.5/src/build/cythonized/sage/structure/parent_old.c:4949) return self.coerce(x) File "sage/structure/parent.pyx", line 1195, in sage.structure.parent.Parent.coerce (/projects/sage/sage-7.5/src/build/cythonized/sage/structure/parent.c:11166) raise TypeError("no canonical coercion from %s to %s" % (parent_c(x), self)) TypeError: no canonical coercion from Full MatrixSpace of 2 by 2 dense matrices over Rational Field to Callable function ring with argument x 
R.<y> = PolynomialRing(QQ)
parent(y)
Univariate Polynomial Ring in y over Rational Field 
A = matrix(QQ, [[1,2],[3,4]])
R.<y> = PolynomialRing(QQ)
f = y^3 - 2*y + 1
f(A)  # works
[ 36 50] [ 75 111] 
show(A.exp())   # compute sum A^k/k!
type(A.exp())
(−122 ((33−11)e33−33−11)e(−12 33+52)233 (33e33−33)e(−12 33+52)111 (33e33−33)e(−12 33+52)122 ((33+11)e33−33+11)e(−12 33+52))\displaystyle \left(\begin{array}{rr} -\frac{1}{22} \, {\left({\left(\sqrt{33} - 11\right)} e^{\sqrt{33}} - \sqrt{33} - 11\right)} e^{\left(-\frac{1}{2} \, \sqrt{33} + \frac{5}{2}\right)} & \frac{2}{33} \, {\left(\sqrt{33} e^{\sqrt{33}} - \sqrt{33}\right)} e^{\left(-\frac{1}{2} \, \sqrt{33} + \frac{5}{2}\right)} \\ \frac{1}{11} \, {\left(\sqrt{33} e^{\sqrt{33}} - \sqrt{33}\right)} e^{\left(-\frac{1}{2} \, \sqrt{33} + \frac{5}{2}\right)} & \frac{1}{22} \, {\left({\left(\sqrt{33} + 11\right)} e^{\sqrt{33}} - \sqrt{33} + 11\right)} e^{\left(-\frac{1}{2} \, \sqrt{33} + \frac{5}{2}\right)} \end{array}\right)
<type 'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense'> 
print "A.inverse()"
A.inverse()
print "A^(-1)"
A^(-1)
print "~A"
~A
print "A.transpose()"
A.transpose()
print "A.conjugate() -- boring since real"
A.conjugate()
print "A.antitranspose()"
A.antitranspose?
print "A.adjoint() -- matrix ov adjoint matrices"
A.adjoint()
A.inverse() [ -2 1] [ 3/2 -1/2] A^(-1) [ -2 1] [ 3/2 -1/2] ~A [ -2 1] [ 3/2 -1/2] A.transpose() [1 3] [2 4] A.conjugate() -- boring since real [1 2] [3 4] A.antitranspose() 
File: /projects/sage/sage-7.5/src/sage/matrix/matrix_rational_dense.pyx
Signature : A.antitranspose()
Docstring :
Returns the antitranspose of self, without changing self.

EXAMPLES:

   sage: A = matrix(QQ,2,3,range(6))
   sage: type(A)
   <type 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'>
   sage: A.antitranspose()
   [5 2]
   [4 1]
   [3 0]
   sage: A
   [0 1 2]
   [3 4 5]

   sage: A.subdivide(1,2); A
   [0 1|2]
   [---+-]
   [3 4|5]
   sage: A.antitranspose()
   [5|2]
   [-+-]
   [4|1]
   [3|0]
A.adjoint() -- matrix ov adjoint matrices [ 4 -2] [-3 1] 
A = matrix(QQ, 3, [1..9])
show(A)
V = A.image()
W = A.kernel()
print "image: "
print V
print "kernel: "
print W
(123456789)\displaystyle \left(\begin{array}{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right)
image: Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] kernel: Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 -2 1] 
A.restrict(V) # V = image
[-6 -6] [18 21] 
A.restrict(W)
[0] 
show(factor(A.charpoly()))
x⋅(x2−15x−18)\displaystyle x \cdot (x^{2} - 15 x - 18)
show(factor(A.restrict(V).charpoly()))
(x2−15x−18)\displaystyle (x^{2} - 15 x - 18)
M = span(QQ, [vector([1,0,0])])   # A doesn't leave the subspace M invariant... so get an error :-)
M
A.restrict(M)
Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 0 0]
Error in lines 3-3 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/matrix/matrix2.pyx", line 4894, in sage.matrix.matrix2.Matrix.restrict (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix2.c:36811) raise ArithmeticError, "subspace is not invariant under matrix" ArithmeticError: subspace is not invariant under matrix

And some fun...

matrix_plot(random_matrix(RDF,50)^3, cmap='summer', colorbar=True)

show(DiGraph(A))
d3-based renderer not yet implemented
n = 10; list_plot3d(random_matrix(RDF, n),
                    texture='red',
                    frame_aspect_ratio=[4, 4, 10], mesh=1)
3D rendering not yet implemented
show(DiGraph(matrix(ZZ,2,2,[2,1,  0,1])))
d3-based renderer not yet implemented
v = vector([1,2,3])
w = vector([0,2,5])

%var s, t
g = parametric_plot3d(s*v + t*w, (s, -1,1), (t,-1,1), opacity=.5)

v = vector([1,-2,3])
w = vector([0,3,-5])
h = parametric_plot3d(s*v + t*w, (s, -1,1), (t,-1,1), color='red')

#i = parametric_plot3d(...., (s, -2, 2), ...)
#g+h+i
show(g+h)
3D rendering not yet implemented