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2.2 Matrix Multiplication - PowerPoint PPT Presentation

2.2 Matrix Multiplication. 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.

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2.2 Matrix Multiplication

• 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?

• 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

• 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

• Given matrices A and B, find the results of the following if possible:

• A2

• B2

• AB

• BA

• 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.

• 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!

• 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:

• 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 

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

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

• 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.

• 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.

• 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. 

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

• 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)

• 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:

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

• We can partition a matrix into smaller blocks:

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

• Go through ex. 11 with finding powers of mtx:

• Find A8

• This was convenient since we have a 0 matrix

• This was convenient since we had a diagonal matrix

• Trace: sum of elements on main diagonal of square mtx

• Idempotent: square mtx,P, is idempotent if P2 = P