What is the important part of a matrix? This was the opening question in Professor Strang’s 2016 Alan Tayler Lecture, given at St Catherine's College, Oxford on Monday 28th November. More specifically, if you have a rectangular matrix of real or complex numbers, can you approximate it with another matrix of lower rank and keep most of the information stored in the original matrix? If all the rows in are the same, then all information is in one of the rows and the entire matrix can be obtained by multiplying the row (from the left) with a vector of ones.
The answer to Professor Strang’s question is given by the Singular Value Decomposition theorem, which provides a splitting of every rectangular matrix into the product of a unitary matrix, a diagonal matrix of non-negative real numbers, and another unitary matrix. The k largest entries in the diagonal matrix defines an approximation to the original matrix where the error is given by the k+1’th largest entry of the diagonal matrix.
Why is Singular Value Decomposition important? Datasets are matrices, and Singular Value Decomposition can help determine the important part of your dataset. A widely used method is Principal Component Analysis, which uses Singular Value Decomposition to find the principal components. In the last part of the lecture, Professor Strang used Principal Component Analysis to analyse how the average grades of all Oxford graduates have increased over the last ten years.