All Questions
Tagged with matrix-decomposition diagonalization
76
questions
20
votes
3
answers
3k
views
Why is there not a test for diagonalizability of a matrix
Let $A$ be square.This question is a bit opinion based, unless there is a technical answer.I think it is helpful tho. Also this question is closely related to this question : quick way to check if a ...
- 477
1
vote
0
answers
49
views
Estimating the parameters of an ellipse
Problem definition
Consider a dataset composed by $m$ bivariate measurements
\begin{equation*}
y_j \sim \mathcal{U}(\mathcal{E}(\ell_1,\ell_2,\theta)) \qquad j=1,2,\dots,m
\end{equation*}
uniformly ...
- 729
2
votes
1
answer
190
views
How to block diagonalize a orthogonal matrix?
I am trying to block diagonalize a real orthogonal matrix, A. The condition is that the blocks should also be orthogonal. I found this pretty old yet abstract paper that says
"By block ...
- 71
0
votes
0
answers
20
views
Discussion about diagonalizing matrix
I have equation
$I = \rho(r)^{T} A_{h} \rho(r)$,
where
$\rho(r)= \begin{pmatrix}\rho_{+}(r) \\ \rho_{-}(r) \\ \rho_{z}(r)\end{pmatrix}$.
I want to rearrange $I$ by diagonalizing $A_{h}$ such as
$\...
- 29
0
votes
1
answer
83
views
SVD of product of diagonal and unitary matrices
Given two (possibly rectangular) diagonal matrices $\Sigma_\text{L}$ and $\Sigma_\text{R}$ with nonnegative elements, what can we say about the singular value decomposition of
$$\Sigma_\text{L} X \...
- 73
0
votes
1
answer
77
views
Simultaneous diagonalization of a positive semi-definite matrix and a symmetric matrix
Let $A,B$ be $d\times d$ symmetric matrices with real coefficients such that $A$ is positive semi-definite. Prove that $\det(tA+B)\in \mathbb{R}[t]$ is either $0$ or has all roots real.
Here is what ...
- 769
1
vote
1
answer
67
views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (https://doi.org/10.48550/arXiv.2006.16364), which seemed to be well-founded. For commuting matrices $\mathbf{A}$ ...
- 528
1
vote
1
answer
691
views
Can we use the singular value decomposition to compute the matrix exponential for a non-diagonalisable matrix?
For a diagonalisable matrix $ \bf{A} $ with eigendecomposition $ \bf{A} = \bf{U} \bf{\Lambda} \bf{U}^{-1} $, we know that $ \exp(\bf{A}) = \bf{U} (\exp \bf{\Lambda}) \bf{U}^{-1} $, where $ \exp \bf{\...
- 332
2
votes
1
answer
92
views
How to diagonalize this (close-to-diagonal) matrix fast?
I have a symmetric matrix that looks like this. (Note that they are not the normal tridiagonal matrices, the diagonals are one diagonal apart). It would be a fairly large matrix. What would be some ...
- 303
8
votes
3
answers
272
views
Find the $n$th power of a 3-by-3 circulant matrix
Consider the matrix given as $$A=\begin{bmatrix}a_0 & a_2 & a_1\\ a_1 & a_0 & a_2\\ a_2 & a_1 & a_0\end{bmatrix}$$
Write a formula for $A^n$ for $n\in\mathbb{N}$.
$$$$
My ...
- 161
0
votes
1
answer
41
views
Relationship between rows and columns of an orthogonal matrix in a spectral decomposition
I am trying to derive the relation $\mathbf A=\mathbf V^T\boldsymbol\Lambda\mathbf V=\sum_{i=1}^n\lambda_i\mathbf v_i\mathbf v_i^T$, where $\mathbf A$ is symmetric, $\mathbf V$ orthogonal (where $\...
- 21
0
votes
1
answer
79
views
Can a certain matrix $A$ be diagonalized by a rotation matrix? [closed]
I have matrix $$\mathcal{A}=\pmatrix{a & c \\ c & b}$$ I can diagonalize this matrix by $\mathcal{A} = SDS^{-1} \Rightarrow D=S^{-1}\mathcal{A}S$, where $D$ is diagonalized matrix consists of ...
- 29
2
votes
0
answers
93
views
Conditions under which a non-symmetric block matrix is diagonalisable
I have a $(2n)\times (2n)$ matrix defined in blocks:
$$
\begin{equation}
\begin{split}
M=\left[
\begin{array}{c|c}
A & B\\
\hline
C & D \\
\end{array}
\right]
\end{split}
\end{equation}
$$
...
- 299
0
votes
1
answer
63
views
One matrix is diagnolized by orthonormal basis of another matrix
Let $A\in \mathbb R^{n\times n}$. Suppose $B$ is symmetric and positive definite, and
\begin{equation}\label{eq:sym}
A^TB=BA,
\end{equation}
then $A$ is diagnolizable by a B-orthonormal basis. ...
- 1
0
votes
1
answer
47
views
Question about inverse of a matrix.
Consider the block upper triangular matrix
$$A = \left[ \begin{matrix} A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} \right], $$
where $A\in\mathbb R^{n\times n}$ and $A_{11}\in\mathbb R^{k\times ...
- 744