Articles of matrices

Determine cycle from adjacency matrix

Is there a way/algorithm to determine if there is a cycle in a graph if I only have the adjacency matrix and can not visualize the graph?

Texture mapping from a camera image (knowing the camera pose)

I’m not sure if I should ask this question here or on stackoverflow, so forgive me if I’m wrong. I want to apply a texture (taken from a camera) on a 3D surface, let me explain my problem: I have acquire a 3D surface through a Kinect camera. At some intervals I have saved the […]

An inequality on the root of matrix products (part 2 – the reverse case)

Suppose $A$ and $B$ are positive definite (symmetric) real matrices. In a previous post (An inequality on the root of matrix products) I asked whether $(AB)^{1/2}+(BA)^{1/2} \geq A^{1/2}SB^{1/2}+B^{1/2}S^TA^{1/2}$ ? where $S$ is a square contractive matrix (i.e. a square matrix that obeys $I-SS^T\geq 0$). This was shown by counterexample to be false in some cases […]

Determinant of $n\times n$ matrix

How can I calculate the determinant of the following $n\times n$ matrix, where $n$ is a multiple of $3$ ? $$\begin{pmatrix} 0 & 0 & 1 & & & & & & &\\ & & & 0 & 0 & 1 & & & &\\ & & & & & &\ddots\\ & & & & […]

What is the fastest numeric method for determinant calculation?

I have a C++ matrix class which can do the following operations on a square matrix related to determinant calculation: LU Decomposition Calculation of eigenvalues Calculation of determinant by adjoint method I want to calculate determinant. Which these there methods is faster? I can say that the answer is not “3”, but at least can […]

If $X$ is symmetric, show $k(X^2)$ = $k(X)^2$

Suppose we have a symmetric invertible matrix $X$. How can I find $k(X^2)$ in terms of $k(X)$? Note that $k$ represents the condition number operation, that is $k(X) = \|X\|\,\|X^{-1}\|$. So, after messing around in Matlab for a while, I think $$k(X^2) = k(X)^2$$ although I dont know how to prove it. I think I […]

Two basic examples of trace diagrams?

In the wikipedia entry on Trace Diagrams (see http://en.wikipedia.org/wiki/Trace_diagram), the statement is made that “The simplest trace diagrams represent the trace and determinant of a matrix”. Could anyone provide me with the graphic representation of those 2 cases for, say, a 3×3 quadratic matrix with any arbitrary number entries? Thanks in advance.

Bounds on off-diagonal entries of a correlation matrix

Assume that all the entries of an $n \times n$ correlation matrix which are not on the main diagonal are equal to $q$. Find upper and lower bounds on the possible values of $q$. I know that the matrix should be positive semidefinite but how to proceed to get the upper and lower bounds? Thanks!

How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$

This question already has an answer here: Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} – \det M_{ij}\det M_{ji}$ 1 answer

Given $XX^\top=A$, solving for $X$

Not equal to this (my) own question. It’s more general, probably more easy than the original question. All of the elements of $X$ and $A$ are integers. $XX^\top=A$ and $A$ is a symmetric matrix. How to find all possible $X$ matrices? Maybe a Gram-Schmidt method to keep only integer solutions. An example: $$ XX^\top= \left( […]