Transcript Math and Operators

Modified Gary Larson Far Side
Matlab
Linear Algebra Review
Matrices can represent sets of
equations!
a11x1+a12x2+…+a1nxn=b1
a21x1+a22x2+…+a2nxn=b2
…
am1x1+am2x2+…+amnxn=bm
What’s the matrix representation?
x1
b1 
a11 a12 ... a1n 
 
 


x2
b2 
a21 a22 ... a2n 




x
b
A


 . 
.


...
 
 


xn


bm 
am1 am2 ... amn 
Ax  b
Vectors
Matrices
U = [umn] =
u11
u21
u12
u22
…
…
u1n
u2n
…
um1
um2
…
umn
The general matrix consists of m rows and n columns. It is also
known as an m x n (read m by n) array.
Each individual number, uij, of the array is called the element
Elements uij where i=j is called the principal diagonal
Transpose of a Matrix
Matrix & Vector Addition
Vector/Matrix addition is associative and commutative
(A + B) + C = A + (B + C); A + B = B + A
Matrix and Vector Subtraction
Vector/Matrix subtraction is also associative and commutative
(A - B) - C = A - (B - C); A - B = B - A
Matrix and Vector Scaling
ax
X
• For addition and subtraction, the size of the
matrices must be the same
Amn + Bmn = Cmn
• For scalar multiplication,
the size of Amn does not matter
• All three of these operations do not differ
from their ordinary number counterparts
• The operators work element-by-element
through the array, aij+bij=cij
Vector Multiplication
• The inner product or dot product
v
a
w

v  w  (x1, x2 ) (y1, y2 )  x1y1  x2 y2
The inner product of vector multiplication
is a SCALAR
v  w  (x1, x2 ) (y1, y2 ) || v || || w || cosa
Inner product represents a row matrix multiplied by a
column matrix. A row matrix can be multiplied by a
column matrix, in that order, only if they each have
the same number of elements!
In MATLAB, in order to properly calculate
the dot product of two vectors use
>>sum(a.*b)
element by element multiplication (.*)
sum the results
A . prior to the * or / indicates
that matlab should perform the array or
element by element calculation rather
than linear algebra equivalent
or
>>a’*b
• The outer product
A column vector multiplied by a row vector.
In Matlab:
>>a*b’
ans =
The outer product of vector
multiplication
is a MATRIX
24 6 30
8 2 10
-12 -3 -15
Matrix Multiplication
Cn p  Anm Bm p
The inner numbers have to match
m
cij   aik bkj
k 1
Two matrices can be multiplied together
if and only if
the number of columns in the first equals
the number of rows in the second.
B
a21
C
A
a22
In MATLAB, the * symbol represents
matrix multiplication :
>>A=B*C
• Matrix multiplication is not commutative!
Ann Bnn  Bnn Ann
C*B
#taken from previous slide
ans =
19 18 17
16 24 32
32 24 16
• Matrix multiplication is distributive and associative
A(B+C) = AB + BC
(AB)C = A(BC)
Revisit the vector example
>>a'*b
>>a*b’
ans =
ans =
11
a1x3 * b3x1 = c1x1
24 6 30
8 2 10
-12 -3 -15
a3x1 * b1x3 = c3x3
Dot products in Matlab
(using this form – built in functions - don’t have to match dimensions of vectors – can mix
column and row vectors – although they have to be the same length)
>> a=[1 2 3];
>> b=[4 5 6];
>> c=dot(a,b)
c =
32
>> d=dot(a,b’)
d =
32
Dot products using built-in function
For matrices – does dot product of columns.
Essentially treating the columns like vectors.
The matrices have to be the same size.
>> a=[1 2;3 4]
a =
1
2
3
4
>> b=[5 6;7 8]
b =
5
6
7
8
>> dot(a,b)
ans =
26
44
Determinant of a Matrix
or follow the diagonals
det (A) = a11a22a33+ a12a23a31 + a13a21a32 − a11a23a32 − a12a21a33 −
a13a22a31

Cross-product of two vectors
The cross product a × b is defined as a vector c that is
perpendicular to both a and b, with a direction given by the
right-hand rule and a magnitude equal to the area of the
parallelogram that the vectors span.
a  b  a b sin 
Cross products in Matlab
(using this form – built in functions - don’t have to match dimensions of vectors – can mix
column and row vectors – although they have to be the same length)
>> a=[1 2 3];
>> b=[4 5 6];
>> e=cross(a,b)
e =
-3
6
-3
>> f=cross(a,b’)
f =
-3
6
-3
>> g=cross(b,a)
g =
3
-6
3
For matrix – does cross product of columns.
(one of the dimensions has to be 3 and takes other dimension as additional vectors)
>> a=[1 2;3 4;5 6]
a =
1
2
3
4
5
6
>> b=[7 8;9 10;11 12]
b =
7
8
9
10
11
12
>> cross(a,b)
ans =
-12
-12
24
24
-12
-12
Matrix Operators
•
•
•
•
•
•
•
•
+ Addition
- Subtraction
* Multiplication
/ Division
\ Left division
^ Power
' Complex conjugate transpose
( ) Specify evaluation order
Array Operators
•
•
•
•
+ Addition
- Subtraction
.* Element-by-element multiplication
./ Element-by-element division.
• A./B: divides A by B by element
• .\ Element-by-element left division
• A.\B divides B by A by element
• .^ Element-by-element power
• .' Unconjugated array transpose
• the signs of imaginary numbers are not changed, unlike a
regular matrix transpose
Operators as built-in commands
plus
- Plus
uplus
- Unary plus
minus
- Minus
uminus
- Unary minus
mtimes - Matrix multiply
times
- Array multiply
mpower
- Matrix power
power
- Array power
mldivide - Backslash or left matrix divide
mrdivide - Slash or right matrix divide
ldivide - Left array divide
rdivide - Right array divide
cross - cross product
+
+
*
.*
^
.^
\
/
.\
./
Multiplication in Matlab
>> x=[1 2];
>> y=[3 4];
>> z=x*y’
z=
11
Regular matrix multiplication – in this case
with vectors 1x2 * 2x1 = 1x1 => dot product
>> w=x.*y
w=
3 8
Element by element multiplication
>> z=x'*y
z=
3 4
6 8
Regular matrix multiplication – in this case
with vectors 2x1 * 1x2 = 2x2 matrix
Division in Matlab
In ordinary math, division (a/b) can be thought of as a*1/b or a*b-1.
A unique inverse matrix of B, B-1, only potentially exists if B is square.
And matrix multiplication is not communicative, unlike ordinary
multiplication.
There really is no such thing as matrix division in any simple sense.
/
:
B/A is roughly the same as B*inv(A). A and B must have
the same number of columns for right division.
\ :
If A is a square matrix, A\B is roughly the same as
inv(A)*B, except it is computed in a different
way. A and B must have the same number of
rows for left division.
If A is an m-by-n matrix (not square) and B is a matrix
of m rows, AX=B is solved by least squares.
The / and \ are related
B/A = (A'\B')’
Inverse of a Matrix
the principal diagonal elements switch
the off diagonal elements change sign
the determinant
• Square matrices with inverses are said to be
nonsingular
• Not all square matrices have an inverse.
These are said to be singular.
• Square matrices with determinants = 0 are
also singular.
• Rectangular matrices are always singular.
Right- and Left- Inverse
If a matrix G exists such that GA = I, than G is a
left-inverse of A
If a matrix H exists such that AH = I, than H is a
right-inverse of A
Rectangular matrices may have right- or leftinverses, but they are still singular.
Some Special Matrices
• Square matrix: m (# rows) = n (# columns)
• Symmetric matrix: subset of square matrices
where AT = A
• Diagonal matrix: subset of square matrices where
elements off the principal diagonal are zero, aij =
0 if i ≠ j
• Identity or unit matrix: special diagonal matrix
where all principal diagonal elements are 1
Linear Dependence
a
b
c
2a + 1b = c
Linear Independence
a
b
c
There is no simple, linear equation that can make
these vectors related.
Rank of a matrix
Acknowledgement
• This lecture borrows heavily from online
lectures/ppt files posted by
• David Jacobs at Univ. of Maryland
• Tim Marks at UCSD
• Joseph Bradley at Carnegie Mellon