(produces one cell at a time)
(produces one column at a time)
(produces one row at a time)
(produces one matrix at a time)
(diagonal entries 1 off-diagonal entries 0)
(identity matrix with rows exchanged)
(Hint: what's the matrix called?)
(subtracting 2 times row 2 from row 3)
(why did we get this? was there something special about C and B?)
Ok, but why was the product a square symmetric matrix?
Hence, if C is the transpose of B,CB will always be a square symmetric matrix
Hence, if C is the transpose of B,CB will always be a square symmetric matrix
(we will now use this result)
(subtracting 2 times row 2 from row 3)
(add 2 times row 2 to row 3 - the inverse of what E1 did)
Notes: the 3d example from the previous slide a 2d example 1 1 0 1 1 -1 0 1
(switch rows 1 and 2)
(switch rows 1 and 2)
(switch rows 1 and 2 again)
(Notes: switched co-ods)
(lower triangular matrix)
(lower triangular matrix)
(rotation matrix)
Trivia: That is a Markov matrix. One of its eigen values is 1.
The corresponding eigen vector is a vector of all 1's
Associative
Distributive
(Not)Commutative
matrix
vector
vector
(distributive property)
(and their connection to matrix multiplication)
Hint: Ax is the linear combination of the columns of A