# 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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### Gram schmidt swapping two vectors

The question has background here but it's really just a linear algebra question. Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process (with no normalization of vector) ...
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### Why is $A^{-1}$ existing a necessary condition for $x$ to be unique in $Ax = b$? [duplicate]

Consider the equation $Ax = b$ where $A \in \mathbb{R}^{m \times n}$, and $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. Say we know $A$ and $b$. I am wondering why, in order for us to uniquely ...
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### I want to bound the entries of the matrix $\mid A^{-1} \mid \mid A \mid$

I want to show that the entries of the matrix multiplication $\lvert A^{-1}\rvert \lvert A \rvert$, where $A\in \mathbb R^{n \times n}$ are less than one. The absolute value in $\lvert A \rvert$ is ...
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### Defining majorization of vectors using doubly stochastic matrix

On Wikipedia, it says that given two non-increasing vectors $a,b\in\mathbb{R}^d$, $a$ majorizes $b$ if and only if there exist a doubly stochastic matrix $D\in\mathbb{R}^{d\times d}$ such that $b=Da$. ...
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### Finding a basis of a Subspace

I have a subspace $U = \langle x^2-x+4,x-1,x^2+x \rangle$ of $P_2$ over $\mathbb R$. I need to find a basis of $U$. We know already that these $3$ vectors span $U$ so we need to check for linear ...
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### Specific value for an invertible matrix

I have been given the following matrix, and have been tasked with finding the values of "a" that makes it invertible. I know that for a matrix to be invertible, then the determinant of said ...
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### Perspective and warp transformations

I'm looking for how to map a rectangular image to perspective and warp transformation. An example of how it works can be found here: https://mockover.com/editor/ So the input is the 4 corners of the ...
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### What does it mean when a system of Linear Equations have more than one solution?

Consider 3 linear equations where one is a linear combination of other two(which are not parallel). Say $a$, $b$ and $a+b$. Now $a+b$ is also a line right? Then how $a$, $b$ and $a+b$ can have more ...
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### How to prove that a matrix smallest eigenvalue is positively correlated with some of its elements?

A block matrix： $$\bf{J}= \begin{bmatrix} (\bf{A}-\bf{Q}) & (\bf{P}-\bf{K})\\ \bf{P} & (\bf{A}+\bf{Q}) \end{bmatrix},$$ where $\bf{A}$ is an $n$-dim positive definite real matrix, and its non-...
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### Eigenvalues of a 8x8 Matrix

This is the 6th problem of the TACA in August, 2023. One have 2 hours to solve 15 problems. I am wondering how to calculate the eigenvalues of the following 8 by 8 matrix by hand. Note that this is ...
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### Integer Linear Programming - Dividing n people into m groups of specific sizes

I've recently asked this question about dividing n people into m groups for the specific model I used to solve the assignment problem of dividing the people into groups (boolean variables xij that ...
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### Properties of matrices with a multiplicative order

I have been trying to find any article or sources talking about the structure and properties of matrices with a multiplicativw order, i.e. a matrix $A$ has a multiplicative order of $n$ if and only if ...
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