All Questions
Tagged with matrix-decomposition linear-transformations
83
questions
3
votes
1
answer
54
views
Completely non- normal matrix
Let $M_n(\mathbb{C})$ be the space of $n \times n$ matrices with complex entries. A matrix $N$ is said to be normal if $N^*N=NN^*$ where $N^*$ denotes the conjugate transpose of $N$. One can think of ...
- 973
1
vote
0
answers
33
views
Euclidean norm of a vector resulting from a matrix multiplied by an orthonormal matrix multiplied by an arbitrary vector!
Let $A \in \{0,1\}^{m\times n}$ be a binary matrix with $ m < n$, and let $U \in \mathbb{C}^{n \times n}$ be an arbitrary orthonormal matrix. Let $\sigma_{min}$ denote the smallest non-zero ...
0
votes
0
answers
26
views
Decomposing a complex square matrix into multiple square matrices
I am working on a problem in optics (called Multi-plane light conversion) where I am trying to find the decomposition of a target transformation (matrix) $\hat{T}\in\mathbb{C}^{(NxN)}$. The form of ...
- 1
0
votes
1
answer
90
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How unique is the orthogonal diagonalization of a real symmetric matrix, if we don't change the diagonal matrix of eigenvalues (no permutaiton)?
Let $A$ be a real symmetric $n\times n$ matrix, so it's orthogonally diagonalizable, i.e. there is $P\in O(n)$ so that $P^{-1}AP=P^{T}AP=D$(diagonal).
I'm asking myself: how unique can $P$ be? Here ...
- 2,364
1
vote
0
answers
36
views
How to parameterize matrices $A$ and $B$ s.t. $AB^T$ is symmetric
Question
I would like to know the most general and optimal form that the $n\times n$ square matrices $A$ and $B$ must take s.t., the matrix $AB^T$ is symmetric. (e.g., $A\in GL(n)$, $B=\lambda I$ ...
- 108
0
votes
1
answer
81
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Null Space of Matrix (Linear Algebra) [closed]
I have the problem in understanding some particular case of Null Space Matrix
Given (3x3 Matrix) RREF (A) =
$$\pmatrix{0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0}$$
Why the vector of ...
0
votes
0
answers
35
views
First step of SVD changes basis of domain to basis such that...
I am trying to understand Singular Value Decomposition. I have not read this explicitly, but as far as I understand it we have:
$\mathbf{V}^\top$ changes the basis from the canonical/standard basis ...
- 67
0
votes
0
answers
28
views
Dualization of matrix inequalities
I am studying Linear Matrix Inequalities (LMI) in Control Theory with this lecture notes:
https://www.imng.uni-stuttgart.de/mst/files/LectureNotes.pdf
I am at the Dualization Section (4.4.1, page 106)
...
2
votes
2
answers
733
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Why do we rotate twice in SVD decomposition
I have read the related discussion and some text and get some basic ideas.
Geometrical interpretations of SVD
Visualization of Singular Value decomposition of a Symmetric Matrix
What I don't ...
- 465
0
votes
1
answer
336
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Proof of existence Schur decomposition
I was reading proof of existence of Schur decomposition.
I understand everything except one thing.
Why submatrices like $A_2$ have same eigen value with main matrix $A$ like $\lambda_2$ ?
A ...
3
votes
1
answer
139
views
Normal Form of Linear Maps Between Matrices
I am looking for a reference which features the following result -- for lack a better term I will call this "normal form" (of a linear map between matrices) -- and which explores its ...
- 4,624
0
votes
1
answer
376
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Prove that if K(kernel matrix) is a positive semi-definite matrix, then k is a dot product: $\exists \phi$ such that $k(x,y) = \phi(x).\phi(y)$
If a kernel matrix is positive semi-definite, how can I prove there exists a $\phi$ s.t $k(x,y) = \phi(x).\phi(y)$
My method:
Every real symmetric matrix can be diagonalized, so we can write:
$$
K = ...
- 345
2
votes
0
answers
149
views
How can I multiply a linear transformation by a antilinear transformation
Well, I have the following inquiry: I have an element $g \in \mathrm{O}(V)$. I know that such element can be decomposed as the sum of two operators: $$g=p_g+q_g,$$ where $p_g$ is linear and $q_g$ is ...
- 2,566
0
votes
0
answers
45
views
Linear transformation on eigenvalues within an eigenvalue decomposition
Suppose I have a diagonalizable real matrix $A$ with the right eigenvectors $X$ and diagonal eigenvalue matrix $\Lambda$ such that $$A = X \Lambda X^{-1}.$$ (In fact in my case $A$ is symmetric so $X^{...
- 43
3
votes
1
answer
194
views
Could the product of a skew-symmetric matrix and an invertible matrix be nilpotent?
Suppose that $A$ is a $d\times d$ skew-symmetric matrix, $B$ is a $d\times d$ invertible matrix and $AB$ is a nilpotent matrix. The unique example I could find is $A=O$, the null matrix.
My question: ...
- 141