Problem of the week - A simple question about eigenvalues
February 20, 2019
One of the most used matrix decompositions is the eigendecomposition, which is related to the concept of diagonalization, and decomposes a matrix into eigenvectors and eigenvalues. Eigendecomposition plays a key role in computer vision and machine learning in general. Well known examples are PCA (Principal Component Analysis) for dimensionality reduction or EigenFaces for face recognition. As another important example of the use of this decomposition, Google, relies upon eigenvalues and eigenvectors to rank pages with respect to relevance.
Although this problem of the week doesn't request to find the eigendecomposition of a matrix, it is related to the concept of eigenvalues, and to solve it, you will need to know the foundation of the procedures used to compute them.
Problem of the week - Gram-Schmidt process
February 6, 2019
Orthogonal bases have some practical advantages and are very useful when dealing with projections onto subspaces. These bases are defined in spaces equipped with an inner product also called a dot product, and by definition, a basis is called orthogonal if every pair of basis vectors are orthogonal, that is, their inner product is 0. When the length of each vector is 1 (vectors are normalized), the basis is called an orthonormal basis.
In an inner product space, it is always possible to get an orthonormal basis starting from any basis, by using the Gram-Schmidt algorithm. To solve this problem of the week you will need to prove you master the Gram–Schmidt process, and you also need to compute the change of basis matrix.
Problem of the week - Column space of a matrix
January 30, 2019
In Linear Algebra the most important subspaces are tied to matrices. One of these subspaces is the Column Space, which consists of all linear combinations of the columns of a matrix. This subspace, spanned by the columns of a matrix, is crucial in Linear Algebra and it is related to 4 of the most important subjects: Matrices, System of Linear Equations, Vector Spaces and Linear Transformations.
This problem of the week is about this subspace, but you should also apply the concept of symmetric matrix.
Problem of the week - LU decomposition
January 9, 2019
An LU decomposition (or factorization) of a matrix A is the product of a lower triangular matrix L and an upper triangular matrix U that is equal to A. One of the motivation for an LU decomposition is the fact that this decomposition can be used as an alternative method to solve systems of linear equations, where once the matrix of the system has been decomposed, the solution of the system can be obtained by solving two easy systems, one by the method of forward substitution and the other by the method of backward substitution. The LU decomposition is another approach designed to exploit triangular systems.
Although is very common to be asked to find an LU decomposition for a square matrix, the concepts are extended to rectangular matrices as well. In this problem of the week, you should deal with the LU decomposition for a rectangular matrix.
