• The triangle inequality $$||x+y||\leq ||x|| + ||y||$$ is possibly the earliest inequality that we learn. Starting from high school, we learn that the shortest distance between two points is a single straight line, and if we want to travel the same distance by two or more straight lines, we will travel longer. We know very well that the triangle inequality that we so intuitively get is not limited only to the 2D or 3D spaces that we can visualize, but to…

• The four fundamental spaces of a matrix $A$, namely the range and the nullspace of itself $A$ or its transpose $A^T$, are the heart of linear algebra. We often find ourselves in need of computing a basis for the range or the nullspace of a matrix, for theoretical or applicational purposes. There are many ways of computing a basis for the range or nullspace for $A$ or $A^T$. Some are better for application, either due to their robustness against floating point…

• By reading this post you’ll be able to comprehend the basic mechanism behind the proof of LU decomposition Schur Inversion Lemma The Sherman-Morrison-Woodbury inversion formula Small perturbations can’t reduce rank (p216-217 of Meyer) Rank of a matrix equals the rank of its largest nonsingular submatrix (p214 of Meyer; see also Exercise 4.5.15) Characteristic polynomial of $AB$ and $BA$ is the same for square $A,B$ (see Exercise 7.1.19, 6.2.16 and eq. (6.2.1) of Meyer) And many more theorems and lemmas (a few…

• Unfortunately a general formula that simplifies the calculations for a matrix inversion $(A+B)^{-1}$ with arbitrary $A$ and $B$ does not exist. If one of the matrices corresponds to a low-rank update (e.g., $B=CD^T$ for some $C, D\in\mathbb{R}^{n\times k}$ with $k<n$), one can use the S.M.W. formula to great effect. However, in other situations, this formula would not simplify calculations. But all hope is not lost yet. There is another case where $(A+B)^{-1}$ can be computed relatively efficiently. To understand how and…

• A matrix inversion formula that frequently appears in machine learning, linear algebra and other textbooks is the Sherman-Morrison-Woodbury formula, according to which a matrix sum can be inverted as $$(A+CD^T)^{-1}=A^{-1}-A^{-1}C(I+D^TA^{-1}C)^{-1}D^TA^{-1}.$$Your first reaction may be wondering if this formula even simplifies anything. The right-hand-side of this equation looks so complicated that one wonders if it’s not simpler to just use compute the sum $A+CD^T$ and invert it. However, this formula can significantly simplify calculations when the matrices $C, D\in \mathbb{R}^{n\times k}$ are…

• In many textbooks of linear algebra you can see that the Schur complement (see below) can be used to invert a non-singular matrix as $$\begin{pmatrix}\mathbf A & \mathbf C\\ \mathbf R & \mathbf B \end{pmatrix}^{-1}=\begin{pmatrix}\mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{CS}^{-1}\mathbf{RA}^{-1} & -\mathbf{A}^{-1}\mathbf{CS}^{-1} \\ \mathbf{-S}^{-1}\mathbf{RA}^{-1}& \mathbf{S}^{-1}\end{pmatrix},$$ where $$\mathbf{S=B-RA}^{-1}\mathbf C$$ is the Schur complement. (We are assuming that the blocks $\mathbf A$ and $\mathbf S$ are both nonsingular.) This formula looks somewhat ugly and, while it is easy to verify (by direct multiplication) that this is indeed the…