Quiz Let a 1 = [1 , 0 , 1] , a 2 = [2 , 1 , 0] , a 3 = [10 , 1 , 2] - PowerPoint PPT Presentation

Quiz Let a 1 = [1 , 0 , − 1] , a 2 = [2 , 1 , 0] , a 3 = [10 , 1 , 2] , a 4 = [0 , 0 , 1]. Compute the row-matrix-by-vector product a 1   a 2    times [7 , 6 , 5]   a 3  a 4 Let v 1 = [2 , 1 , 0 , 1] , v 2 = [4 , − 1 , 0 , 2] , v 3 = [0 , 1 , 0 , 1]. Compute the column-matrix-by-vector product       v 1 v 2 v 3 times [2 , − 1 , 1]      

The Matrix [3] The Matrix Neo: What is the Matrix? Trinity: The answer is out there, Neo, and it’s looking for you, and it will find you if you want it to. The Matrix , 1999

Two views of same process Row-matrix-by-vector multiplication and column-matrix-by-vector multiplication are same!         a 1 b 1 c 1 a 2 b 2 c 2          times [ x 1 , x 2 , x 3 ]   .   .   .   . . .         . . .        a m b m c m       a 1 b 1 c 1 a 2 b 2 c 2       = x 1 + x 2 + x 3  .   .   .  . . .       . . .       a m b m c m       x 1 a 1 x 2 b 1 x 3 c 1 x 1 a 2 x 2 b 2 x 3 c 2       = + +  .   .   .  . . .       . . .       x 1 a m x 2 b m x 3 c m

Two views of same process Row-matrix-by-vector multiplication and column-matrix-by-vector multiplication are same!       a 1 b 1 c 1 a 2 b 2 c 2       = + + x 1 x 2 x 3  .   .   .  . . .       . . .       a m b m c m       x 1 a 1 x 2 b 1 x 3 c 1 x 1 a 2 x 2 b 2 x 3 c 2       = + +  .   .   .  . . .       . . .       x 1 a m x 2 b m x 3 c m   x 1 a 1 + x 2 b 1 + x 3 c 1 x 1 a 2 + x 2 b 2 + x 3 c 2   = .   .   .   x 1 a m + x 2 b m + x 3 c m

Two views of same process Row-matrix-by-vector multiplication and column-matrix-by-vector multiplication are same!       x 1 a 1 x 2 b 1 x 3 c 1 x 1 a 2 x 2 b 2 x 3 c 2       = + +  .   .   .  . . .       . . .       x 1 a m x 2 b m x 3 c m   x 1 a 1 + x 2 b 1 + x 3 c 1 x 1 a 2 + x 2 b 2 + x 3 c 2   =  .  .   .   x 1 a m + x 2 b m + x 3 c m

Two views of same process   x 1 a 1 + x 2 b 1 + x 3 c 1 x 1 a 2 + x 2 b 2 + x 3 c 2   =  .  .   .   x 1 a m + x 2 b m + x 3 c m   ( a 1 , b 1 , c 1 ) · ( x 1 , x 2 , x 3 ) ( a 2 , b 2 , c 2 ) · ( x 1 , x 2 , x 3 )   = .   .   .   ( a m , b m , c m ) · ( x 1 , x 2 , x 3 )  [ a 1 , b 1 , c 1 ]  [ a 2 , b 2 , c 2 ]   =  times [ x 1 , x 2 , x 3 ]  .  .   .  [ a m , b m , c m ]

Two views of same process         a 1 b 1 c 1 a 2 b 2 c 2         We showed  times [ x 1 , x 2 , x 3 ]   .   .   .   . . .         . . .        a m b m c m   [ a 1 , b 1 , c 1 ] [ a 2 , b 2 , c 2 ]   =  times [ x 1 , x 2 , x 3 ]  .  .   .  [ a m , b m , c m ]           [ a 1 , b 1 , c 1 ] a 1 b 1 c 1 a 2 b 2 c 2 [ a 2 , b 2 , c 2 ]           so  is same as   .   .   .    .  . . . .           . . . .          a m b m c m [ a m , b m , c m ]

Row matrix and column matrix are same  1 2 3   1 2 3  4 5 6 4 5 6     Example:  same as     7 8 9 7 8 9    10 11 12 10 11 12 Same underlying math’l object, different representations ◮ column-list representation ◮ row-list representation of a MATRIX One operation, matrix-vector multiplication, with two interpretations: ◮ dot-product interpretation: output vector entries are dot-products of rows with input vector ◮ linear-combinations interpretation: output vector is linear combination of columns where coeff’s are input vector entries You must memorize which is which.

The Matrix Traditional notion of a matrix: two-dimensional array. � 1 � 2 3 10 20 30 ◮ Two rows: [1 , 2 , 3] and [10 , 20 , 30]. ◮ Three columns: [1 , 10], [2 , 20], and [3 , 30]. ◮ A 2 × 3 matrix. For a matrix A , the i , j element of A ◮ is the element in row i , column j ◮ is traditionally written A i , j ◮ but we will use A [ i , j ]

List of row-lists, list of column-lists ◮ One obvious Python representation for a matrix: a list of row-lists: � 1 � 2 3 represented by [[1,2,3],[10,20,30]] . 10 20 30 ◮ Another: a list of column-lists: � 1 � 2 3 represented by [[1,10],[2,20],[3,30]] . 10 20 30

List of row-lists, list of column-lists Ungraded “Quiz”: Write a nested comprehension whose value is list-of- row -list representation of a 3 × 4 matrix all of whose elements are zero:   0 0 0 0 0 0 0 0   0 0 0 0 Hint: first write a comprehension for a typical row, then use that expression in a comprehension for the list of lists. Answer: >>> [[0 for j in range(4)] for i in range(3)] [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]

The matrix revealed The Matrix Revisited (excerpt) http://xkcd.com/566/ Definition: For finite sets R and C , an R × C matrix over F is a function from R × C to F . ◮ R = { a , b } and C = { @ , # , ? } . @ # ? a 1 2 3 ◮ R is set of row labels b 10 20 30 ◮ C is set of column labels In Python, the function is represented by a dictionary: {(’a’,’@’):1, (’a’,’#’):2, (’a’, ’?’):3, (’b’, ’@’):10, (’b’, ’#’):20, (’b’,’?’):30}

Rows, columns, and entries @ # ? a 1 2 3 b 10 20 30 Rows and columns are vectors, e.g. ◮ Row ’a’ is the vector Vec( { ’@’, ’#’, ’?’ } , { ’@’:1, ’#’:2, ’?’:3 } ) ◮ Column ’#’ is the vector Vec( { ’a’,’b’ } , { ’a’:2, ’b’:20 } )

Dict-of-rows/dict-of-columns representations @ # ? a 1 2 3 b 10 20 30 One representation: dictionary of rows: {’a’: Vec({’#’, ’@’, ’?’}, {’@’:1, ’#’:2, ’?’:3}), ’b’: Vec({’#’, ’@’, ’?’}, {’@’:10, ’#’:20, ’?’:30})} Another representation: dictionary of columns: {’@’: Vec({’a’,’b’}, {’a’:1, ’b’:10}), ’#’: Vec({’a’,’b’}, {’a’:2, ’b’:20}), ’?’: Vec({’a’,’b’}, {’a’:3, ’b’:30})}

Our Python implementation @ # ? a 1 2 3 b 10 20 30 >>> M=Mat(({’a’,’b’}, {’@’, ’#’, ’?’}), {(’a’,’@’):1, (’a’,’#’):2,(’a’,’?’):3, (’b’,’@’):10, (’b’,’#’):20, (’b’,’?’):30}) A class with two fields: ◮ D , a pair ( R , C ) of sets. ◮ f , a dictionary representing a function that maps pairs ( r , c ) ∈ R × C to field elements. We will later add lots of matrix class Mat: operations to this class. def __init__(self, labels, function): Example: For a Mat M , M [ r , c ] self.D = labels is the entry in row r , column c . self.f = function

Identity matrix For any domain D , there is a matrix that represents the D -to- D identity function f ( x ) = x a b c ------- a | 1 0 0 b | 0 1 0 c | 0 0 1 Definition: D × D identity matrix is the matrix 1 D such that 1 D [ k , k ] = 1 for all k ∈ D and zero elsewhere. Usually we omit the subscript when D is clear from the context. Often letter I (for “identity”) is used instead of 1 Mat(({’a’,’b’,’c’},{’a’,’b’,’c’}),{(’a’,’a’):1,(’b’,’b’):1,(’c’,’c’):1}) Quiz: Write procedure identity( D ) that returns the D × D identity matrix over R represented as an instance of Mat . Answer: def identity(D): return Mat((D,D), (k,k):1 for k in D)

Converting between representations Converting an instance of Mat to a column-dictionary representation: @ # ? 1 2 3 a b 10 20 30 Mat(({’a’,’b’}, {’@’, ’#’, ’?’}), {(’a’,’@’):1, (’a’,’#’):2, (’a’,’?’):3, (’b’,’@’):10, (’b’,’#’):20, (’b’,’?’):30}) ⇒ {’@’: Vec({’a’,’b’}, {’a’:1, ’b’:10}), ’#’: Vec({’a’,’b’}, {’a’:2, ’b’:20}), ’?’: Vec({’a’,’b’}, {’a’:3, ’b’:30})} Quiz: Write the procedure mat2coldict(A) that, given an instance of Mat , returns the column-dictionary representation of the same matrix. Answer: def mat2coldict(A): return {c:Vec(A.D[0],{r:A[r,c] for r in A.D[0]}) for c in A.D[1]}