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.
57,489 questions
1 vote
0 answers
48 views
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 ...
- 11
5 votes
0 answers
84 views
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)...
- 597
-2 votes
0 answers
23 views
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 ...
0 votes
1 answer
68 views
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, $$ \frac{\partial f}{\partial x_1} = \begin{bmatrix}\frac{\...
- 355
2 votes
1 answer
61 views
Scattering formula in terms of matrix operations
I have a transformation between two $2 \times 2$ matrices, given by $$ \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} = \frac{1}{T_{22}} \begin{pmatrix} -T_{21} ...
- 1,236
2 votes
1 answer
104 views
$P$ 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 $R=c_1 P+c_2 Q$ ...
- 1,385
1 vote
0 answers
29 views
Constructive method for expressing arbitrary matrix permutations using cyclic row/column shifts
I am unsure which category this question best fits into, so I apologize in advance if this is not the ideal place to ask. It is known (see for example: Do cyclic permutations of rows and column ...
- 11
2 votes
0 answers
44 views
An alternating trilinear form on $2\times 2$ traceless matrices
While pondering why the Levi-Civita symbol shows up in the commutation relations for Pauli matrices, I found that $\langle A,B,C\rangle=\text{tr}(ABC)$ is an alternating trilinear form on traceless $2\...
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