# Tagged Questions

For questions about matrices which are defined block wise, like $\pmatrix{A&B\\ C&D}$ where $A,B,C$ and $D$ are themselves matrices. Use this tag with (matrices), and often with (linear-algebra).

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### Partition of a Matrix

In Linear Algebra, we have been taught that the partition of a matrix $A$ consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to ...
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### Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
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### Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
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### Properties about Matrices that can be proved by only using Block Multiplication of Matrices

I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove ...
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### Determinant, number of non zero columns

Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty : Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...
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### Computations for LDA: Eigendecomposition

While reading the book Elements of Statistical Learning p. 113, the author used eigendecomposition of the covariance matrix $\hat{\Sigma}_k =\mathbf{U}_k\mathbf{D}_k\mathbf{U}_k^T$ where ...
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### Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK))$$ $$K = 0:dK:\pi$$ Where K is a ...
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### Is there a name for the following type of block matrices?

Is there a name for the following type of block matrices? A matrix $A$ is [insert name here] if it can be decomposed into non-zero non-scalar submatrices such that each sub-matrix $B$, with $B$ ...
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### Condition for a block matrix to be positive semi-definite

Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? ...
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### Solving Toeplitz plus Diagonal System, how should I make use of the structure?

I learned that a Toeplitz system, $Ax = B$ where $A$ is Teoplitz, can be solved in $O(n \log n)$ time using Superfast method. or approximate $A$ similar to Approximation method. I am keep ...
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### Computing characteristic polynomial of tridiagonal block matrices

I want to compute the characteristic polynomial of symmetric matrices of the form \begin{bmatrix} A & U & & & 0\\ U & B & V & &\\ & V& C& W &\\ ...