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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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 ...
3
votes
1
answer
95
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Why is fact that the determinant of this matrix is 0 equivalent to $x'^T Fx=0$
I am reading Multiple View Geometry in Computer Vision and in the chapter 17.1, it talks about the following matrix which needs to have 0ドル$ determinant.
$$
\begin{bmatrix}
A & x & \textbf{0} \\...
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can a pivotal variable be equal to zero using gauss elimination method, even if you have your AX= B where b is defined to be 1/2 [closed]
Find the solution of the following linear system using the Gauss elimination method:
x1 + 2x2 − x3 + x4 = 3,
2x1 − x2 + x3 + x5 = 2,
3x1 + x2 + x4 − x5 = 4
The particular solution is same as chatgpt ...
2
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0
answers
56
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Most general Logarithm of a matrix
Consider any matrix $A \in \text{GL}_d(\mathbb{C}),ドル i.e, a square invertible matrix. We define a logarithm of $A$ as any matrix $X$ such that $$e^X = A.$$
Our objective is to find of possible ...
3
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2
answers
227
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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},ドル 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 ...
5
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86
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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)...
2
votes
1
answer
42
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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 ...
0
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1
answer
77
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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),ドル then by the chain rule,
$$
\frac{\partial f}{\partial x_1}
=
\begin{bmatrix}\frac{\...
0
votes
1
answer
26
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Bounds for $\|R(P+Q+R)^{-1}\|_2$?
I have three symmetric positive semidefinite matrices $P,Q,R$ where $P$ is strictly positive definite. I have $L := R(P+Q+R)^{-1}$. I need a way to find a bound
$$ x^{\top} L^{\top} L x \le \kappa_P x^...
2
votes
1
answer
62
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Scattering formula in terms of matrix operations
I have a transformation between two 2ドル \times 2$ matrices, given by
$$
\begin{pmatrix}
S_{11} & S_{12} \\
S_{21} & S_{22}
\end{pmatrix}
=
\frac{1}{T_{22}}
\begin{pmatrix}
-T_{21} ...
1
vote
1
answer
44
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A few clarifications about multiplication in subgradient calculus
In the subgradient calculus linearity properties, the appropriate side of the addition rule utilizes Minkowski addition of sets. Ordinarily in linearity, a scaling rule agrees with, and is basically ...
2
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1
answer
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$P$ is a good matrix for $A$ if $P^{-1} A P=A^T$....
Let $P,Q,R,A$ are $n\times n$ non-singular matrices and $A^T$ denotes transpose of $A$. If $$P^{-1}AP=A^T,\quad Q^{-1}AQ=A^T$$
Then $P$ and $Q$ are good matrices for $A$.
$(1)$ Prove $R=c_1 P+c_2 Q$ ...
0
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1
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Can a 2D linear transformation have a finite set of invariant lines?
I've been trying to prove that if a 2D linear transformation has an invariant line through the origin then it must have an infinite set of invariant lines that are parallel to this line. I've searched ...