# 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.

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### Matrix notation for double sum of row and column with the same index

a biologist/ecologist here, I never took any courses in algebra (definitely missing in my education) but I am working with matrices all day. For one paper, I have to write the matrix formulation but ...
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### How to show that the matrix $R^TCR$ is diagonal if $R$ is a rotation matrix related to $C$ in a specific way?

I have two matrices: $C=\begin{bmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{bmatrix}$ and $R=\begin{bmatrix}\cos\theta &-\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$. I would like to ...
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### Struggling to prove that the elements of an inverse matrix satisfy a certain equation.

We have three vectors $\vec{e_1},\vec{e_2},\vec{e_3}$ that are not necessarily orthogonal or normalised, but do form a basis. We also have a matrix $G$ with elements $G_{ij}=\vec{e_i} \ . \vec{e_j}$,...
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### How to compute the smallest eigenvalue efficiently? [on hold]

$A$ is a $m \times m$ symmetric PSD matrix whose top $n$ eigenvalues are equal to $1$ and whose remaining $(m-n)$ eigenvalues are zero. Here, $n \ll m$. Let $D$ be a diagonal matrix with all diagonal ...
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### Let $A \in \mathbb R^{5 \times 5}$. Prove that $A^{4} + I \neq O$

Let $A \in \mathbb R^{5 \times 5}$. Prove that $$A^{4} + I \neq O$$ I understand that I have to start from $A^{4}+I= 0$ and come up with a contradiction. Is it the Cayley Hamilton I have to use?
### Prove that the determinant is greater than $1$
Let $A$ be an $n \times n$ matrix whose diagonal entries are strictly positive and off-diagonal entries are negative. The sum of the entries on each column is $1$. Prove that $\det(A) > 1$. I ...