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.
57,483 questions
6
votes
2
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Solving linear system with triangular Toeplitz matrix
I'm searching for the solution $\bar{a}$ to the system of equations $\bar{e}_1 = B\bar{a}$ given by
\begin{equation}
\left[\begin{array}
& 1 \\
0 \\
\vdots \\
0 \\
0
\end{...
- 920
2
votes
1
answer
70
views
The opposite category $\mathbf{Mat}_R^\text{op}$
For a commutative ring $K$, the category $\mathbf{Mat}_K$ is defined as follows:
objects are positive integers $m, n, ...$
morphisms $A \colon n \to m$ are $(m \times n)$-matrices, and for two ...
- 2,636
-2
votes
1
answer
37
views
Finding change of basis matrices in $\mathbb{R}^2$ [closed]
In $\mathbb{R}^2$, consider the following ordered basis:
$$B=((1,0),(0,1)),\text{ }C=((-1,1),(1,1)),\text{ and }D=((\sqrt{3},1),(\sqrt{3},-1))$$
Find the change of basis matrix from $B$ to $C$, from $...
- 27
5
votes
2
answers
183
views
Checking if the determinant of a quasi-tridiagonal matrix is nonzero
From a homework assignment for an undergraduate course on numerical methods:
Check if the determinant of the following matrix is nonzero.
\begin{bmatrix}
1 & -2 & 1 & & & &...
- 99
0
votes
1
answer
82
views
This seem to be a sufficient condition for non-commutative matrix product to be symmetric. Is it also necessary? [closed]
I observed that for matrices of following forms, their product are also symmetric:
a b c
b c b
c b a
or perhaps also other similar forms.
Truth to be told, I'm ...
- 355
7
votes
1
answer
124
views
Connection between two methods of finding the inverse of an invertible matrix
I've learned two methods of calculating the inverse of an $n \times n$ non-singular matrix $A$:
By creating an augmented matrix with $I_n$ and using row-reduction to transform the original matrix ...
- 140
3
votes
1
answer
114
views
A generalized Vandermonde matrix appearing in linear recurrence relation
Is the below matrix invertible?
$$ M(\lambda_1,\dots,\lambda_m,n_1,\dots,n_m):= \begin{pmatrix}
1^0 \lambda_1^1 & \cdots & 1^{n_1-1}\lambda_1^1 &\cdots &\cdots &1^0 \lambda_{m}^1 ...
- 1,943
0
votes
1
answer
54
views
Matrices that are related to their transpose via diagonalizable matrices
I'm currently working with matrices having the following property:
Let $A \in M_n(\mathbb Z)$ be square matrix such that there exist diagonalizable matrices $S,T \in M_n(\mathbb C)$ with $A = S A^t T$,...
- 199
-6
votes
0
answers
48
views
The Imitation Number Formula in the Collatz Conjecture is it novel [closed]
I have been analysing the Collatz Conjecture and have identified an infinite family of numbers, which I call 'Imitation Numbers' (N), that share an identical initial trajectory structure with a ...
2
votes
1
answer
140
views
Finding inverse of a $3 \times 3$ matrix with Cayley-Hamilton, and Diophantine equation
I am trying to use this formula to find the inverse of a $3 \times 3$ matrix.
$$ \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}^{...
- 23
1
vote
2
answers
206
views
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 ...
- 5,082
0
votes
0
answers
21
views
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 ...
- 1
3
votes
1
answer
131
views
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 $0$ ...
- 41
4
votes
1
answer
125
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 ...
- 43
3
votes
2
answers
248
views
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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