In [4]:

D=diagm(1:5);
(D,D')

Out[4]:

(
5x5 Array{Int64,2}:
 1  0  0  0  0
 0  2  0  0  0
 0  0  3  0  0
 0  0  0  4  0
 0  0  0  0  5,

5x5 Array{Int64,2}:
 1  0  0  0  0
 0  2  0  0  0
 0  0  3  0  0
 0  0  0  4  0
 0  0  0  0  5)

In [9]:

issym(D)

Out[9]:

true

If S is symmetric, inv(S) ' = inv(S') = inv(S) hence inv(S) is symmetric¶

In [5]:

S=[3 12;12 5]

Out[5]:

2x2 Array{Int64,2}:
  3  12
 12   5

In [8]:

round(inv(S),7)

Out[8]:

2x2 Array{Float64,2}:
 -0.0387597   0.0930233
  0.0930233  -0.0232558

If R is any rectangular matrix R'*R is symmetric: Why?¶

In [12]:

R=rand(5,2); round(R*R',3)

Out[12]:

5x5 Array{Float64,2}:
 0.871  1.166  0.816  0.085  0.987
 1.166  1.598  1.004  0.113  1.35 
 0.816  1.004  0.972  0.081  0.856
 0.085  0.113  0.081  0.008  0.096
 0.987  1.35   0.856  0.096  1.14

In [11]:

issym(R*R')

Out[11]:

true

In [13]:

a=R'*R

Out[13]:

2x2 Array{Float64,2}:
 1.7064   1.93981
 1.93981  2.88402

In []:

Fact without proof: if S is symmetric, elimination produces S=LDL'

Permutation Matrices¶

In [21]:

A=randn(5,5);
(L,U,P)=lu(A);
(round(L,3),round(U,3),round(P,1))

Out[21]:

(
5x5 Array{Float64,2}:
  1.0     0.0     0.0     0.0    0.0
  0.835   1.0     0.0     0.0    0.0
  0.607   0.411   1.0     0.0    0.0
 -0.885   0.216  -0.248   1.0    0.0
 -0.519  -0.577  -0.639  -0.984  1.0,

5x5 Array{Float64,2}:
 -0.831  -1.29    0.041   0.83    2.648
  0.0     2.539   1.151  -0.157  -4.152
  0.0     0.0    -1.898  -1.537   0.432
  0.0     0.0     0.0     0.841   4.13 
  0.0     0.0     0.0     0.0     1.541,

5x5 Array{Float64,2}:
 0.0  1.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0
 0.0  0.0  0.0  0.0  1.0
 1.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  1.0  0.0)

In [22]:

(round(P*A,3),round(L*U,3))

Out[22]:

(
5x5 Array{Float64,2}:
 -0.831  -1.29    0.041   0.83    2.648
 -0.694   1.461   1.186   0.537  -1.941
 -0.505   0.261  -1.4    -1.097   0.332
  0.736   1.689   0.682   0.453   0.784
  0.431  -0.794   0.527  -0.186  -1.78 ,

5x5 Array{Float64,2}:
 -0.831  -1.29    0.041   0.83    2.648
 -0.694   1.461   1.186   0.537  -1.941
 -0.505   0.261  -1.4    -1.097   0.332
  0.736   1.689   0.682   0.453   0.784
  0.431  -0.794   0.527  -0.186  -1.78 )

A real vector space is a set of "vectors" with an addition and multiplication by scalars that satisfies eight rules (rules coming soon)¶

vectors in standard format¶

In [27]:

v=rand(0:9, 5);  w=rand(0:9,5); [v w]

Out[27]:

5x2 Array{Int64,2}:
 4  0
 6  3
 8  9
 3  8
 8  3

In [28]:

v*10

Out[28]:

5-element Array{Int64,1}:
 40
 60
 80
 30
 80

In [29]:

v+w

Out[29]:

5-element Array{Int64,1}:
  4
  9
 17
 11
 11

arrays themselves are vectors¶

In [31]:

A=rand(0:9,2,2); B=rand(0:9,2,2); (A,B)

Out[31]:

(
2x2 Array{Int64,2}:
 3  3
 9  5,

2x2 Array{Int64,2}:
 2  9
 7  7)

In [32]:

A*10

Out[32]:

2x2 Array{Int64,2}:
 30  30
 90  50

In [33]:

A+B

Out[33]:

2x2 Array{Int64,2}:
  5  12
 16  12

functions are vectors¶

In [35]:

typeof(sin)

Out[35]:

Function

In [36]:

sin+cos

no method +(Function,Function)
at In[36]:1

In [37]:

+(f::Function,g::Function)=x->f(x)+g(x);

In [38]:

(sin+cos)(12)

Out[38]:

0.3072810407320572

In [39]:

sin(12)+cos(12)

Out[39]:

0.3072810407320572

In [39]:

*(c::Number,f::Function) = x->c*f(x);

In [40]:

(10*sin)(5)

Out[40]:

-9.589242746631385

In [41]:

10*sin(5)

Out[41]:

-9.589242746631385

In [44]:

x=-10:.001:10; plot(x, (sin+cos)(x));

In [47]:

plot( x, (10cos)(x));