Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
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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 ...
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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 ...
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Showing existence of symplectic transformations preserving a quadratic form
Question: I need help to prove the following statement.
Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...
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need a Jacobian of an orthonormal basis for the column space of a matrix $\mathbf{X}$
I require to find a linear decomposition of any tall matrix $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$, where $M \geq N$, into some uniquely defined decomposition $\mathbf{X}$ $=$ $\mathbf{A}$ $\...
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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 ...
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Exponeintal of symmetric triangular matrix
I want to know the exponeintal of given $n \times n$ symmetirc real tridiagonal matrix ${\bf K}_n$, which is defined as
$${\bf K}_n=\begin{bmatrix}
0 & a & 0 & 0 & \dots & 0 & ...
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Optimization of eigenvalue of matrix with discrete variables
Suppose we have a matrix $H$ like this, where $H_{ii}$ is a discrete variable (for example out of $[1,2,3]$) and $H_{ij}$ is a value that depends on the two adjacent values ($H_{ii}$ and $H_{jj}$). $H$...
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$UDU^{T}$ factorization of a matrix
So, there is this question in which I am suppose to find $UDU^{T}$ factorization of $$A = \begin{bmatrix} -3&1&-1\\1&-3&1\\-1&1&-3 \end{bmatrix}$$
I don't understand what is ...
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Woodbury matrix identity with a minus sign
Is there a form of Woodbury matrix identity
$(A + UCV )^{-1} = A^{-1} - A^{-1}U (C^{-1} + VA^{-1}U )^{-1} VA^{-1}$
But with a minus sign? i.e. $(A - UCV )^{-1}$ It seems like I have to painfully ...
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Proving that the rank of the following matrix is $6$.
In my research work I have come across a matrix which has the rank equals to $6$. I begin defining my problem as follows: Let $P \in \{0,1\}^{7 \times 7}$ denote the right shift matrix defined by
$ P =...
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Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_{min}$, of $A$?
What I want is something like: $\sigma_{min}$...
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Decompose a $3 \times 3$ orthogonal matrix into a product of rotation and reflection matrices
It holds that every orthogonal matrix can be expressed as a product of rotation and reflection matrices.
We can prove that every $2 \times 2$ orthogonal matrix can be represented in the form
$$\begin{...
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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 ...
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Any decomposition of inverse of nonnegative diagonal matrix times a PSD matrix plus lambda times Identity?
I generally have to solve the following system:
$$
(DA + \lambda I)^{-1} v
$$
where $D$ is a diagonal matrix with nonnegative entries, $A$ a symmetric, positive semi-definite (PSD) matrix, $I$ is the ...
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How to obtain this equation of hyperspectral image restoration using total variation in the following equation?
While reading the paper "Total-Variation-Regularized Low-Rank Matrix
Factorization for Hyperspectral Image Restoration" I came through the following equation. Can someone help me how the last equation ...
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