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2.2 Matrix MultiplicationPowerPoint Presentation

2.2 Matrix Multiplication

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Background Example

- We return to our Sweatshirt store example:
- We wish to find the value of the inventory by size
- Smalls - $10, Med - $11, Large - $12, XL - $13

- How would we set up the mult. to do this?

Matrix Multiplication Algorithm

- The mtx mult algorithm is defined to do just that:

- (i,j): multiply entries in row i of first mtx by the corresponding entries in col j of second mtx, and then add terms.
- Note that this is the dot product of row i and col j

A few things to note

- Given the way that the algorithm is defined, what must be true about the dimensions of the matrices in order for multiplication to work?
- # of entries in each row of first matrix must equal # of entries in each column of second matrix
- (i.e. number of columns of first matrix must equal the number of rows of second matrix)
- So if multiplying matrices of following order: (a x b) x (c x d), b = c

- Note: order of solution mtx is:

a x d

Example

- Given matrices A and B, find the results of the following if possible:
- A2
- B2
- AB
- BA

Example#2

- Given the matrices A and B, find AB and BA:

- B is the identity matrix, I3, since AB = BA = A
- The identity is always a square matrix with 1’s on the main diagonal and 0’s elsewhere.
- I is only the identity for a matrix of the same size.

Properties of Mtx Multiplication

- IA = A, BI = B
- A(BC) = (AB)C
- A(B+C) = AB + AC, A(B-C) = AB - AC
- (B+C)A = BA + CA, (B - C)A = BA - CA
- k(AB) = (kA)B = A(kB)
- (AB)T = BTAT
- Note: commutativity does not hold (AB ≠ BA in most cases).
- Therefore the order of the factors in a product of matrices makes a difference!

Helpful Notation

- To help us prove the properties, it is useful to understand the following notation for matrix multiplication.
- A = [aij] is m x n, B = [bij] is n x p
- ith row of A is [ ai1 ai2 …… ain]
- jth column of B is:--------------------->

- ij entry of prod mtx is dot prod of row i of A and col j of B:

Proving Property 3 (also in book)

- Property 3: A(B + C) = AB + AC
- A is m x n, B is n x p, C is n x p
- B + C = [bij + cij]

- This is just the (i,j) entry of AB + the (i,j) entry of AC
- Therefore A(B + C) = AB + AC

Matrix form of linear system

- Try to write the following system as a single matrix equation:

- A is coefficient mtx, X is solution mtx, B is constant mtx

Precursor to Theorem 2

- If X1 is a sol’n to AX=B and X0 is a sol’n to the related homogeneous system AX = 0, then:
- X1 + X0 is a solution to AX = B:

- Theorem 2 is a converse of this.

Theorem 2

- If X1 is a sol’n to AX=B, then every solution, X2, to AX=B is of the form:
- X2 = X1 + X0 where X0 is a solution to the associated homogeneous system AX=0.

Proof of Theorem 2

- If X2 and X1 are both sol’ns to AX=B,
- So AX1 = B and AX2 = B:
- say X0 = X2 - X1 so X2 = X0+X1
- AX0 = A(X2 - X1) = AX2 - AX1= B - B = 0
- Therefore, X0 is a solution to the associated homogeous system AX = 0.

Example

- Express every solution to the following system as the sum of a specific solution plus a solution to the associated homogeneous system.

Solution

- If we take t=0, we get a specific solution, X1
- Therefore, if t≠0, the solution, X2 = X1 + X0 where X0 is a solution to the associated homogeneous stm: AX = 0
- So, gives all sol’ns to assoc. hom. System
- (Show)

Example

- Find row 3 and column 2 of AB.
- In many situations, it is helpful to write a matrix as a column of rows or as a row of columns:

Simplifying Notation

- So then we can use the following notation in mtx mult:

A further simplification

- We can partition a matrix into smaller blocks:

Continuing the Example

- We need to make sure that the way we partition the matrices allows us to multiply matrices which match up appropriately by dimension. (show)

Example continued

- This was convenient since we have a 0 matrix

- This was convenient since we had a diagonal matrix

Other topics in homework

- Trace: sum of elements on main diagonal of square mtx
- Idempotent: square mtx,P, is idempotent if P2 = P

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