4.4.1 Matrix-matrix multiplication: Motivation

Dr. Robert van de Geijn: Currently that
we have matrix vector multiplication cleanly made even away, we'' re ready to
introduce one more procedure– matrix- matrix reproduction. Matrix-matrix multiplication is
most likely the most important procedure. In fact,
later on in the subsection, you'' re visiting that every one of the
various other operations that we'' ve seen before are just unique instances of
matrix-matrix reproduction. To encourage matrix-matrix
reproduction, what we'' re mosting likely to do is go back to the opener for today. We ' re going to chat a little much more concerning our weather forecast design. Recall that we had this table that forecasted the weather for tomorrow based on what the climate was today.And one of the last concerns we asked was,
just how to calculate the table that informs us how to anticipate the weather the day after tomorrow from the weather today? And also we discussed exactly how that was a. issue of tracking exactly how the system basis vectors were transformed by increasing. by the transition matrix that takes a state vector. from eventually to the next. What we ' re mosting likely to now do is associate. this to'matrix-matrix reproduction.
What we have is the state element. for the day after tomorrow. And what we suggested previously. today was that that was a matter of taking. the change matrix and increasing it times. the state element that anticipates tomorrow ' s weather.And that consequently was a matter of.

saying, look, take the state vector that anticipates the weather condition for today, increase. it by the shift matrix that provides you to prediction for tomorrow,. and also after that take the result of that and multiply it by. the shift matrix.
As well as now we get the forecast. for the day after tomorrow. It shouldn ' t be also.
surprising that if we had a table that forecasts the weather condition.
tomorrow from the weather condition today, we need to in a similar way get a table that.
predicts the weather for the day after tomorrow from the weather today.And if there ' s a shift. matrix that takes us from
the state vector for today. to the state vector for tomorrow, there should additionally be a. shift matrix that takes us from today to. the day after tomorrow.
So what does that imply? We must have the ability to feed in. a state vector for day absolutely no.
And we must have the ability to discover a. matrix that then takes that state vector to the forecast. for the day after tomorrow. As well as the action of that matrix should. be the same to first increasing by this transition matrix, P– we. called this P earlier this week– and also
after that taking the result of that. and multiplying it by P too.
What we ' re going to see is that. you can take these two matrices and from them construct. this new matrix, Q. And also the operation that takes these. two matrices right into this brand-new matrix that has the same net effect as. first applying the very first matrix and after that using the 2nd. matrix, that particular procedure we are
going to call. matrix-matrix multiplication.
Allow ' s look at this. differently one more time.What really is going. on below is that we ' re applying some linear. transformation'to the vector x. And afterwards we ' re taking the. outcome of that and applying an additional straight transformation,. which takes place to just coincide direct change. in this certain instance. And what we ' re actually doing. is, we'' re making up
these two linear makeovers. Once again, in this specific instance, we ' re. composing the direct makeover with itself. What we ' re going to see currently is that. there'' s a new vector function that ' s called L sub Q of'x that has the same. internet effect of first applying L sub P to x and after that help applying. L below P to the result of that.And the inquiry, certainly, is, is.

that really a straight improvement? And also if it is, after that we understand that. there must be such a matrix Q? If matrix Q exists, then
. we ' ve seen lot of times now how to in fact calculate.
the columns of that matrix. All you do is, you.
feed in the system basis vectors ahead up with the columns. So in this certain. case, we ' re feeding in the unit basis vector e sub 0.
As well as out pops the very first column. of matrix Q. Similarly, we might calculate the other.
columns of Q. In recap, this triggers some concerns. The very first inquiry is, is LQ of x, which. is defined as a composition of L of P with itself, a straight transformation? The even more basic question is,. if you compose an arbitrary linear transformation with. another straight change, is that a straight transformation? And if it is, exactly how are matrices. An and also B and C related, where A is the matrix. that stands for L sub A, B is the matrix that. stands for L below B, and also C is the matrix that represents this. new direct change, L sub C? So that ' s what we ' ll be talking. regarding for the remainder of the week.

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