Rank and Matrix Multiplication - Advanced Engineering Math - Tutorial Slides, Slides of Engineering Mathematics

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trix ultiplication

1.1. Brief Review—Rank

fi iti :

i atrix , the rank of MMMM is the maximum number of linearly i nt ro s (or columns) in MMMM. r hen e say # of linearly independent rows and # of linearly independent columns, t re essentially the same for a matrix. we will see its proof soon

r :

l t ry ro (or column) operation does not alter the rank. r ll the proof we did in class, we only need to check whether each type of elementary r ( r colu n) operations will decrease the rank or not

t :

i atrix , apply elementary row operations and reduces it to ro echelon form. The rank of MMMM is the number of non-zero rows.

l fi d the rank for following matrixs.

lllleeee 1111

rank=?

rank= 1

rank= 2

l fi d the rank for following matrixs.

lllleeee 1111

if 0 , 2 0 -

a b

a b

a

a

⎡ ⎤ ≠ ⎢^ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦

0 if 0 , 0

b a b

⎡ ⎤ = (^) ⎢ ⎥ ⎣ ⎦

if , rank= 1

otherwise, rank= 2

a = ± b

if =0, rank= 0

otherwise, rank= 2

b

, rank= 2 = 0 , rank= 1 = = 0 , rank= 0

a b a b a b

l sion: ± ≠

r : t be an matrix, then row-rank(A)=column-

r nk( ).

lllleeee 3333

n ×

l t ry ro operation

elementary column operation

r - rank(A)= (^3) column-rank(A)= 3

lllleeee 3333 .... eeeetttthhhhoooodddd 1111

: le entrary row operations will not change column-rank(A).

( )

1 11 21 1 1 1 2 12 22 2 2 2 1 2 1 2

= , , , , , , , = , , , , , , , = , , , , , , , = , , , , , , ,

i k m i k m j j j ij kj mj n n n in kn mn

c a a a a a c a a a a a c a a a a a c a a a a a

⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

11 12 1 1 21 22 2 2

1 2

1 2

1 2

j n j n

i i ij in

k k kj kn

m m mj mn

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎥⎦

⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋮ ⋮ ⋮

t r operation

( )

' 1 11 21 1 1 1 ' 2 12 22 2 2 2 ' 1 2 ' 1 2

= , , , , , , , = , , , , , , , = , , , , , , , = , , , , , , ,

k i m k i m j j j kj ij mj n n n kn in mn

c a a a a a c a a a a a c a a a a a c a a a a a

⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

1 2 ' ' ' 1 2

, , , are linear independent

iff , , , are linear independent

k k

c c c c c c

r : t be an matrix, then row-rank(A)=column-

r nk( ).

lllleeee 3333 .... eeeetttthhhhoooodddd 2222

n ×

t r ( )=r, then by definition of rank, A has r linear independent ic can be denoted by , then all rows of A are li r binations of these,

vvvv 1 (^) , vvvv (^) 2 ,..., vvvv r

1 11 1 12 2 1 2 21 1 22 2 2

1 1 2 2

r r r r

m m m mr r

c c c

c c c

c c c

note that: (^) ( )

( )

1 2

1 2

i i i in

j j j jn

a a a

v v v

aaaa

vvvv

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

11 12 1 11 11 12 1 12 21 22 2 1 1 2 21 22 2 21 11 12 1 22 21 22 2 2 1 2

, ,..., , ,..., , ,..., , ,..., , ,..., , ,..., , ,..., , ,...,

n n n r r r rn n n n r r r rn

a a a c v v v c v v v c v v v a a a c v v v c v v v c v v v

= + + + = + + +

⋯ ⋯ ⋮ ⋮ (^ a m^ 1 ,^ am^ 2 ,...,^ am^ n ) =^ cm^ 1 ( v 1^ 1 ,^ v 1^ 2 ,...,^ v 1^ n ) +^ cm^ 2 ( v 2^ 1 ,^ v 2^ 2 ,...,^ v 2^ n ) +^ + cm^ r ( vr^ 1 ,^ vr^ 2 ,..., vrn )

⋮ ⋮ ⋯

r : t be an matrix, then row-rank(A)=column-

r nk( ).

lllleeee 3333 .... eeeetttthhhhoooodddd 2222

n ×

) ( ) ( ) ( ) ) ( ) ( ) ( )

11 12 1 11 11 12 1 12 21 22 2 1 1 2

21 22 2 21 11 12 1 22 21 22 2 2 1 2

, ,..., , ,..., , ,..., , ,..., , ,..., , ,..., , ,..., , ,...,

n n n r r r rn n n n r r r rn

v v v c v v v c v v v v v v c v v v c v v v

= + + + = + + +

⋯ ⋯ ⋮ ⋮ 1 , m^ n^ v 2 ,. v ..^ m ,^ v^ )^^ =^1 c (^^11 n , v^^ 12 , v ..^ m .,^1 ) v^ +^^2 (^21 , nc^^ 22 ,. v ..^ ,^ v 2^ m ) r^ +^ r v + r (^1 r , n^2 ,..., )

⋮ ⋮ ⋯

1 2

k k

mk

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

k k k r rk

k k k r rk

mk m k m k mr rk

a c v c v c v

a c v c v c v

a c v c v c v

r : t be an matrix, then row-rank(A)=column-

r nk( ).

lllleeee 3333

n ×

f l rollary:

( ) rank(A')

( ) in( ,n)

i r vectors each having n components, if n<p, then these vectors r li rly dependent. If i t square, either the row vectors or the column vectors of A are li rl i dependent

If t r vectors of a square matrix are linearly independent, so are the l v ctors, and conversely.

lllleeee 4444

rt ( badditivity): Let A,B be matrixs, then we

have.

m × n r( AAAA + BBBB ) ≤ r( AAAA ) +r( BBBB )

tr ct a 2 n × matrix

⎡ ⎤ ⎢ ⎥ ⎣ ⎦

AAAA 0000 0000 BBBB

rank =rank( )+rank( )

⎛ (^) ⎡ ⎤⎞ ⎜ (^) ⎢ ⎥⎟ ⎝ ⎣^ ⎦⎠

AAAA 0000 AAAA BBBB 0000 BBBB

tr f r

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0000 AAAA 0000 AAAA AAAA 0000 BBBB AAAA BBBB AAAA AAAA BBBB

t r k of sub atrix is less than or equal to the rank of original matrix, thus:

rank( + ) rank

⎝ ⎣^ ⎦⎠

AAAA 0000

AAAA BBBB

0000 BBBB

(t this, k rows in submatrix are linear independent only if the rr nding k rows in original matrix are linear independent) o about (^) r( AAAA - BBBB ) +r( BBBB----CCCC ) ≥ r( AAAA----CCCC )?