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.
57,486 questions
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Matrices that are related to their transpose via diagonalizable matrices
I'm currently working with matrices having the following property:
Let $A \in M_n(\mathbb Z)$ be square matrix such that there exist diagonalizable matrices $S,T \in M_n(\mathbb C)$ with $A = S A^t T$,...
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The Imitation Number Formula in the Collatz Conjecture is it novel [closed]
I have been analysing the Collatz Conjecture and have identified an infinite family of numbers, which I call 'Imitation Numbers' (N), that share an identical initial trajectory structure with a ...
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Finding inverse of a $3 \times 3$ matrix with Cayley-Hamilton, and Diophantine equation
I am trying to use this formula to find the inverse of a $3 \times 3$ matrix.
$$ \mathbf A^{-1} = \frac{1}{\det(\mathbf A)} \sum_{s=0}^{n-1}\mathbf A^{s} \sum_{k_{1}, k_{2},\dots,k_{n-1}} \prod_{l=1}^{...
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How to symbolically find inverses of matrices? [duplicate]
Today I met the following problem.$\newcommand\b\boldsymbol$
If $\b A$, $\b B$, $\b A+\b B\in\Bbb R^{n\times n}$ are non-singular matrices, find the inverse of $\b A^{-1}+\b B^{-1}$.
The solution is ...
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Graphs where nodes are connected but distant [closed]
Is it possible to make a graph consisting of 2n nodes such that every node is connected to every other node in at least n steps except n of the other nodes?
For example with 1, we make the graph with ...
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Verifying One-vs-All Precision and Recall calculations from a multi-class confusion matrix
I am studying multi-class classification metrics and want to confirm the correct way to compute them from a confusion matrix.
A weather classifier labels days as Sunny, Rainy, Cloudy. The test results ...
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Why is fact that the determinant of this matrix is $0$ equivalent to $x'^T Fx=0$
I am reading Richard Hartley & Andrew Zisserman's Multiple View Geometry in Computer Vision (2nd edition). In section chapter 17.1, it is mentioned that the following matrix needs to have $0$ ...
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can a pivotal variable be equal to zero using gauss elimination method, even if you have your AX= B where b is defined to be 1/2 [closed]
Find the solution of the following linear system using the Gauss elimination method:
x1 + 2x2 − x3 + x4 = 3,
2x1 − x2 + x3 + x5 = 2,
3x1 + x2 + x4 − x5 = 4
The particular solution is same as chatgpt ...
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Most general Logarithm of a matrix
Consider any matrix $A \in \text{GL}_d(\mathbb{C})$, i.e, a square invertible matrix. We define a logarithm of $A$ as any matrix $X$ such that $$e^X = A.$$
Our objective is to find of possible ...
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Showing the Cayley transform sends positive definite matrices to small matrices and vice versa
Given a matrix $Z\in\Bbb R^{n\times n}$, write $Z\succ0$ to mean that $\langle v,Zv\rangle>0$ when $v\ne0$. (We may say that $Z$ is positive definite, but note that $Z$ is not required to be ...
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Limit as $x\to 0$ of $Z(x)=\bigl[B(x I-A)C+D(x I+A)^{-1}E\big]^{-1} $
Let $A,B,C,D,E$ be $n\times n$ complex matrices. Assume that $B,C,D,E$ are invertible, and that $A$ is singular (non-invertible). Consider the matrix-valued function
\begin{equation}
Z(x)=\Bigl[B(xI-A)...
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Upper-bounded version of the Gale-Ryser theorem
The standard Gale-Ryser theorem is for the existence of a $(0,1)$-matrix given exact row sums $R = (r_1, \dots, r_K)$ and exact column sums $C = (c_1, \dots, c_M)$. What if we relax the column sums ...
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How to Interpret Detection Errors and Compute Performance Metrics for an Autonomous Pedestrian Detection System? [duplicate]
I am analyzing the performance of an autonomous vehicle’s pedestrian detection system, and I want to ensure that I am interpreting the scenario correctly in terms of confusion-matrix components. This ...
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Notation Convention in Linear Models
Notation Convention in Linear Models: Why $\theta^\top x$ instead of $\theta x$?
Question:
I'm working with CMU 10-414 Lecture 2 and I'm curious about the notation convention used to represent the ...
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Second Derivative Chain Rule in Matrix Notation
If we have a function $f(x_1,x_2,x_3,x_4)$ and perform a coordinate transformation to $f(y_1,y_2,y_3,y_4)$, then by the chain rule,
$$
\frac{\partial f}{\partial x_1}
=
\begin{bmatrix}\frac{\...
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