All Questions
Tagged with matrix-decomposition matrices
1,732
questions
0
votes
0
answers
30
views
How to estimate the inverse of a non-invertible matrix?
So I'm working on a machine learning problem where my solution requires taking the inverse of a matrix at some point. The problem is that this matrix is sometimes non-invertible. In theory the the ...
- 11
0
votes
2
answers
51
views
For any SVD $A = U\Sigma V^T$ of a positive definite, symmetric matrix $A \in \mathbb{R}^{n \times n}$, we have $U = V$.
First of all, I've read through all of the answers here and here, but neither of those threads was able to give completely satisfying answers.
Now, I understand that, if $A$ is symmetric and positive ...
- 23
4
votes
0
answers
47
views
Relationship between BCH code and asymmetric Ramanujan bipartite graph ( possibility for a research collaboration)
I have been working on a research topic that deals with the binary matrices arising from the BCH codes by selecting code vectors of specific weight while discarding the rest of the code vectors that ...
- 49
0
votes
0
answers
28
views
Woodbury matrix identity with a minus sign
Is there a form of Woodbury matrix identity
$(A + UCV )^{-1} = A^{-1} - A^{-1}U (C^{-1} + VA^{-1}U )^{-1} VA^{-1}$
But with a minus sign? i.e. $(A - UCV )^{-1}$ It seems like I have to painfully ...
- 1,634
0
votes
0
answers
50
views
Proving that the rank of the following matrix is $6$.
In my research work I have come across a matrix which has the rank equals to $6$. I begin defining my problem as follows: Let $P \in \{0,1\}^{7 \times 7}$ denote the right shift matrix defined by
$ P =...
- 49
0
votes
0
answers
27
views
Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_{min}$, of $A$?
What I want is something like: $\sigma_{min}$...
- 115
0
votes
1
answer
27
views
Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?
I know in general, matrix multiplication is not commutative, but would it be true in this special case?
$D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
- 149
-2
votes
1
answer
81
views
What is the square root of a square matrix squared? [closed]
Admittedly, made the title a little funny, but this is a valid question.
I have come across the following equation
$$
I x^2=AA
$$
where $I$ is a unit matrix, $A$ is a square matrix of the same ...
- 101
1
vote
0
answers
23
views
Compressed image using SVD draws a clear line between part that's blank and part with a drawing. Why? [closed]
I'm trying to compress grayscale images using SVD. This is the original image:
Yes, there's a lot of blank space.
I then choose the x% largest singular values, perform the transformed matrices ...
0
votes
1
answer
23
views
For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?
Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows:
$$...
- 155
0
votes
0
answers
25
views
Singular values on streching the vectors
For the following statement: for a vector $x$ and a matrix $A$, if the vector $x$ is not in the null space of $A$, the vector $x$ will at least be stretched by the smallest non-zero singular value, i....
- 115
2
votes
0
answers
26
views
For a symmetric matrix $B$ and following four relevant matrices $P,Q,C,D$, what's the relation between $QP$ and $CDC^{\rm T}$?
Suppose $B\in\mathbb{R}^{n\times n}$ is a symmetric matrix with ${\rm rank}(B)=r$. Then $B$ is equivalent to $\tilde{B}$ in (1), where $I_{r}$ denotes the identity matrix of order $r$. That is, there ...
- 155
0
votes
0
answers
72
views
Find one quartic root of a matrix
I have found the previous spectral decomposition of the matrix $$A=\begin{pmatrix} 1 & 1 & 0 \\
0&1&1\\
1&0&1
\end{pmatrix}.$$
You can see I verified such decomposition indeed ...
- 1,494
0
votes
0
answers
20
views
2x2 blocks in the QZ algorithm
How are the $2\times2$ blocks supposed to be diagonalized in the QZ-Algorithm? Taking the matrix pencil (A,B) and finding it's generalized Hessenberg decomposition (H,R) for which $\exists Q,Z \in \...
0
votes
1
answer
45
views
Matlab qz algorithm not reliable
I programmed my own version of the qz algorithm and, while testing it's results using the matlab qz algorithm, I found a particular case where my solution reaches an upper-triangular matrix and matlab ...