# 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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### Is there a matrix that can be used to find the transpose of a matrix?

Let $A$ be a general $n\times n$ invertible matrix. Let $T^A$ be the "transposer" matrix i.e. $T^A A = A'$. (Does that $T^A$ multiplied by $A$ equal the transpose of $A$?) Then does $T^A$ depend on ...
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1answer
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### How to find the dimension of the intersection of hyperplanes?

Let P ⊂ R4 be the subspace with equation 2x + 3y − z + w = 0, let Q ⊂ R4 be the hyper-plane with equation x + y −z = 0, let R ⊂ R4 be the hyper-plane with equation x−y + w = 0, and let l = P∩Q∩R. I ...
1answer
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### Problem on a set of all $3 \times 3$ real upper triangular matrices with all diagonal entries $=1$,

Let $W$ be the set of all $3 \times 3$ real upper triangular matrices with diagonal entries $1$ and let $B = (b_{ij})$ be a $3 \times 3$ real matrix trhat satisfies $AB = BA$ for all $A \in W$ then ...
1answer
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### How much shift Eigenvalues?

Let $A$ be an $n × n$ matrix. Then, we can create a family of matrices $A(t) = tB + D$ where $D$ is the same as $A$ with all the off-diagonal entries reduced to zero and $B$ is the same as $A$ with ...
0answers
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### Union of graphs is a sort of matrix concatenation

Just an intuition for discussion: If we consider a set $G=\lbrace G_1, G_2\rbrace$ of two connected simple graphs, where $G_1$ and $G_2$ have no vertex in common then the the graph union $G_1\cup G_2$...
1answer
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### Projection Matrix Formulae Comparison and Intuition

I wanted some intuition behind the formulae of projection of point to a subspace. Particularly I wanted to compare it to the situation where the subspace is just a 1D line. Let $b$ be the point to ...
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### Proving that a Matrix has an Inverse using a Polynomial

Suppose an $(n × n)$ matrix $A$ satisfies that an equation such as, $A^{2} − 3A + I = 0,$ where $I$ is the $(n × n)$ identity-matrix, prove that A has an inverse. Any tips on approaching a proof?
2answers
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### Linear Functions View as Surfaces or Vectors

I am familiar(on a basic level) with concepts of inner product, dual space and linear functionals. Reviewing my understanding of simple linear functions of the type $Y=ax+by+cz$ I see that this can be ...
2answers
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### Is there an easy way to prove the matrix $(XYZ)^{-1} = Z^{-1}Y^{-1}X^{-1}$? [closed]

I've been using pro numerals to show this relationship, and it is taking ages to complete because of all the steps necessary! Any other easier way to prove this statement?
0answers
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### Complexity of calculating $(A^n)_{i,j}$

I'm looking for the most efficient way to calculate a single entry of a matrix that was $n$ times multiplied with itself . Pet example and motivation: Let us denote the Stirling numbers of the second ...
0answers
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### Minimal set of $n\times n$ matrices whose products generate all $n\times n$ matrices with a single $1$

Let $n \in \mathbb{N}^*$ be the dimension of the matrices. Let $M_{i,j}$ be the $n\times n$ matrix with $1$ at position $(i,j)$ and 0 elsewhere. Let $S_n$ be the set containing all the $n^2$ ...
2answers
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