Eun Hyun Jedd2 (Matrix Multplication)

7/27/2019 Eun Hyun Jedd2 (Matrix Multplication)

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Matrix Multiplication

Hyun Lee, Eun Kim, Jedd Hakimi


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2.1 Matrix Operations

Key IdeaMatrix multiplication corresponds to

composition of linear transformation.

The definition of AB, is critical for the

development of theory and application.

Then what is the definition of AB?


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What is AB?

The subscripts tell the location of an entry.If A is m x n matrix, then m represents

the row and n represents the column. Inthe product AB, left-multiplication by Aacts on the columns of while right

multiplication by B actions on the rows ofA. In other words..


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Definition of AB continues..

Columns j Columns j

of AB = A x of B

Also, the following is true.

(row i of AB)=(row i of A) x B


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How do we add matrix?

Let me show you an example:suppose

A= 2 3 B= 4 77 8 6 5 ,

then what is A+B?

= 2+4 7+3 = 6 10

7+6 8+5 13 13


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Example of Bx

X= 1 1 13 2 5 then.. What is 4x?

4x= 4 4 4

12 8 20


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Matrix Multiplication

How do we multiply the matrix?When matrix x is multiply by B, we are transforming x

into vector Bx. Lets say we multiply by Bx by A again,now we have created A(Bx). This is the key concept tomove on to next steps!


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We can always express A(Bx) and (AB)x to

represent this composite mapping as a matrixthat was multiplied by a single factor.


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Example

IfAis a 4 x 5 matrix and Bis a 5 x 3matrix, what are the sizes ofABand BA, if

they are defined?


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Remember! You cant multiply anymatrix. There are some conditions!The number of columns on the first matrix

has to be same as the number of rows of

the second matrix.


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Theorem 1

A,B, and C are the matrices of same size and let r and s

be the scalars. A+B= B+A 1.1 (A+B)+C=A+(B+C) 1.2 A+0=A 1.3

r(A+b)=rA+rB 1.4 (r+s)A= rA+sA 1.5 R(sA)=(rs)A 1.6


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Lets prove it!

Note that all the vectors of A, B, and C hasthe same size. Lets say 2nd column of vectorof A, B, C is A2, B2, and C2(respectively).

Then 3(A2+B2)= 3A2+3B2 The first matrix has the same size as the

matrix on the right: The corresponding

columns are equal so from 1.1-1.6 it can beproved using same logic as I proved 1.4


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A X B

A= 3 0 B= 4 71 1 6 8

5 2

A x B=3(4)+0(6), 3(7)+0(8)

1(4)+1(6), 1(7)+1(8)

5(4)+2(6), 5(7)+2(8)= 12 21

10 15

32 51


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Row-Column Rule

If we think of the ith row of A and the jthcolumn of B as vectors, then theelement in the ith row and the jth columnof C is, scalar product of the ith row of Aand jth column of B.

(AB)ij=Ai1Bij+Ai2Bi2+.+AinBnj

Rowi(AB)=rowi(a) * B


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Theorem 2

A, B, and Cs are matrices they have sizesfor which the indicates sums andproducts are defined

A(BC)=(AB)C 2.1

A(B+C)=AB+AC 2.2

(B+C)A=BA+CA 2.3r(AB)=(rA)B=A(rB) 2.4

ImA=A=AIn 2.5


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Lets prove it

A(BC)=(AB)C ------Associative lawFirst observe that both A(BC) and

(AB)C are m x n matrices.

Let Uj denote column j of AB.

Since Uj==ABj, column j of A(BC)isAUj= A(BCj).

Further more, column j of (AB)C

is (AB)Cj=A(Bcj).It follows that the corresponding columns

of A(BC) and (AB)C are equal.


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Example of Associative Law

A=(1,2), B= 3 4 C= 3 0 22 1 5 1 0

AB=(1,2) 3 4 = (7,6)2 1

(AB)C=(7,6) 3 0 2 =(51,6,4)

5 1 0A(BC)=(1,2) 29 4 6 =(51,6,14)

11 1 4


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Proof of Distributive Law

Both (A+B)C and AC +BC are m *n matrices, andso we compare the corresponding columns ofeach matrix. For any j, ACj and BCj are the jthcolumns of AC and BC, respectively. But ..

*jth column of (A+B)C is (A+B)Cj=ACj+BCj.

By property of matrix jth column of AC+BC is

equal to the above. This left distributive lawapplies to right distributive law, too.


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Example of Distributive Law

A=(1,2) B= 3 4 C= 4 2

0 5 1 7

B+C= 7 6 A(B+C)=(9,30)

1 12

AB=(3,14)

AC=(6,16)

AB+AC=(9,30)


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AB is not equal to BA

A = 1 B= (3,4,1,5); AB= 3 4 1 52 6 8 2 10

0 0 0 0 0

1 3 4 1 5

BA

=(3,4,1,5) 1 = (3+8+5)=16

2

0

1


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The Transpose of a Matrix

Sometimes it is of interest to interchange therows and columns of a matrix

The transpose of a matrix A=Aij is a matrixformed from A by inter changing rows andcolumns such that row i of A becomes columns Iof the transpose matrix. The transpose is

denoted by At and At=Aji when A= Aij


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Example of the Transpose

A= 1 3 At= 1 22 5 3 5

A= 1 3 4 At= 1 00 1 0 3 1

4 0

It will be observed that ifAis m x n,At isn x m


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Matrix Powers

If matrix A is squared it isdenoted as ..

=A2

Ak=A x A x A xA

A0=1(its theconventionright??)

Therefore, A0

X=X (itself)Also, you we can apply that

(Ap)x(Aq)= A(p+q)


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Theorem 3

Suppose A and B represent matrices

of appropriate sizes for the following

sums and products.


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Theorems (a)-(c) are obvious,

so the proofs are not required.

For theorem (d),


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2.4 Partitioned Matrices

I see youve chosen the Red Pill

Welcome to

Part i t ioned Matr ices

-Not as exciting as an action movie,

but it might still make you sayWhoa.


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By now youre teeming with anticipation,eager to expand your mind. Lets start

by clearing a few things up by

understanding what we are working with.

Why dont I ask your first question for you

(because you might feel silly talking to a

computer screen):


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Q. What is a Part i t ioned Matr ixand what

does it have to do with me?

A. Ah, good question.

Well, a Part i t ioned Matr ixis a matrix that hasbeen broken down into several smaller matrices

But why tell you when I can show you a picture.

Lets say I have a 5x4 Matrix called G


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And now a partitioned version (with the partitionlines in red):


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And now we name the individual parts

(AKA: BlocksorSubmatr ices):


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Now we can rewrite G as a 3x2 Matrix:

Now doesnt that look a lot nicer than our original G?

Of course is does.

Now, to address the second part of your question,

this partitioned matrix can help us by speeding upa supercomputer or something like that. Isnt that exciting?

Dont answer that. On to more questions


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Q. Can I addPart i t ioned Matr ices

to each other?

A. Sure, as long as the matrices being multiplied

are identical in the way they are partitioned.Each b lockcan be added to its corresponding

block.

Too easy. How bout another question?


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Q. Can I multiply Part i t ioned Matr ices

by a scalar?

A. Sure, as long you multiply the scalar

one block at a time.

Come on, give me a harder one


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Q. How can I multiply Part i t ioned Matr icesby each other?

A. Okay, now youve asked a tough one.

The best way to explain this is through anexample. In the following example uppercase

Letters will represent blocks.

First matrix J partitioned like so:


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Or

And Matrix K partitioned like so:


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Or


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First of all, Part i t ionedMatrices can

Only be multiplied because the numberof vertical partitions of the first matrix

in the equation is equal to the number

of horizontal partitions of the second

matrix in the equation.

matrix:

(Row 1:) AE+BF

(Row 2:) CE+DF


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Now we expand each one of the Blocks.


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Q. What Theorems can we get from this

method of multiplying PartitionedMatrices?

A. Well youre a curious one arent you?

Well, in fact there is a theorem. Its called theColumn-row Expansion Theorem and it basicallymeans that because blocks work so well formultiplication then columns of the first matrixin a multiplication equation will correspond torows of the second matrix in that equation tomake an easier way to compute the equation.Heres the proof:


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For each row index Iand column indexj,

the (i,j)- entry in column n(A) row n(B) isthe product of a(i,N)from column n(A)and b (N, j)from row n(B). Hence the(i, j)entry in the sum shown in (1) is:

a (i, 1)b(1,j)+a (i, 2)b(2,j)++a (i,N)b(N,j)This sum is also the (i, j)-entry in the AB, by therow-column rule.

Or you can just take my word for it.


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The End

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Eun Hyun Jedd2 (Matrix Multplication)