There may be some different sub-collections of columns from that are linearly dependent, but every collection of columns is guaranteed to be linearly dependent. So if the column rank of is, then there is some sub-collection of columns of which are linearly independent. The column rank of a matrix is the size of the largest possible subset of ‘s columns which are linearly independent. If are not linearly independent, then they’re linearly dependent. A collection of vectors is linearly independent if there is no linear combination of them which produces the zero vector, except for the trivial -weighted linear combination. A linear combination of vectors is a weighted sum of the form, where are scalars 2 In our case, matrices will be comprised of real numbers, making scalars real numbers as well. At the core of linear algebra is the notion of a linear combination. Let’s do a quick review of the foundations of linear algebra. How can we compress a low-rank matrix? Can we use this compressed matrix in computations? How good of a low-rank approximation can we find? What even is the rank of a matrix? What is Rank?
#NONMEM S MATRIX ALGORITHMICALLY SINGULAR HOW TO#
But there any many questions about how to use this tool and how widely applicable it is. This example is suggestive that low-rank approximation, where we approximate a general matrix by one of much lower rank, could be a powerful tool. The rank of the matrix is whereas almost certainly possesses the maximum possible rank of. The answer is that the matrix has low rank. This leads us naturally to the following question: Linear algebraically, in what way is simpler than ? (1) and our previous discussion that is much more efficient to store than. When presented in this linear algebraic form, it’s less obvious in what way is simpler than, but we know from Eq. Let us call the matrix on the right-hand side of Eq. (1) corresponds to the matrix approximation
The entry corresponding to station and day is the temperature at station on day. Let’s collect our weather data into a matrix with 1000 rows, one for each station, and 365 columns, one for each day of the year. Let us abstract this approximation procedure in a linear algebraic way.
#NONMEM S MATRIX ALGORITHMICALLY SINGULAR FULL#
Further, we have massively compressed our data, only needing to store the numbers rather than our full data set of 365,000 temperature values. However, we might plausibly expect it to be fairly informative. This model is clearly grossly inexact: The weather does not satisfy a simple sinusoidal model. If we are particularly bold, we might conjecture that the weather approximately experiences a sinusoidal variation over the course of the year:įor a station, denotes the average temperature of the station and denotes the maximum deviation above or below this station, signed so that it is warmer than average in the Northern hemisphere during June-August and colder-than-average in the Southern hemisphere during these months. Temperatures are correlated across space and time: If it’s hot in Arizona today, it’s likely it was warm in Utah yesterday. However, we have reasons to believe that significant compression is possible. If we were to store each of the temperature measurements individually, we would need to store 365,000 numbers. 1 I borrow the idea for the weather example from Candes and Plan. Suppose that there are 1000 weather stations spread across the world, and we record the temperature during each of the 365 days in a year. Let’s start our discussion of low-rank matrices with an application.