# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

35,069 questions
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### Geometric justification of a rotation matrix

From S.L Linear Algebra: We can define a rotation in terms of matrices. Indeed, we call a linear map $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a rotation if its associated matrix can be ...
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### algebraic expression of Matrix product [on hold]

Suppose $M = X^T \Delta X$, where $X$ and $\Delta$ are $P \times P$ matrices and $\Delta$ is symmetric. Can anyone give a simple algebraic expression of the matrix $M$?
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### The Effect of Adding an Edge on the Laplacian of a Weighted Digraph

Let $G$ be a weighted digraph with Laplacian $L:=D-A$, where $D$ is the degree matrix and $A$ is the incidence matrix. Is there any result on the behavior of the eigenvalues of $L$ when we add an edge ...
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### Find all units in the ring Z[i] = { a+bi : a,b ϵ Z } [duplicate]

Find all units in the ring $Z[i]$= { $a+bi$ : $a,b$ ϵ $Z$ }. I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($\pm1$). I ...
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### minimise the trace of a matrix over all column permutations

I have a 10x10 positive symmetric matrix, I need to find the optimal permutation of the columns in order to minimise the trace. I can't try all permutations because that would be a 10! problem. Any ...
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### Show that $\exp(C^{-1} AC ) = C^{-1} \exp(A C)$
Show that $\exp(C^{-1} AC) = C^{-1} \exp(A C)$ for any matrices $A \in L_{n}(\mathbb{R})$ and $C \in GL_{n}(\mathbb{R})$. The hint of the question is given below: Consider the linear operator \$\...