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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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Singular Value Decomposition of a Real Unit Matrix

Given a real matrix $A \in \mathbb{R}^{m \times n}$ whose entries are all ones, what is the reduced/short/economy singular value decomposition $A = U\Sigma V^T$? I can see that we have a single ...
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12 views

Is Frobenius norm constant?

I came across an article that makes me doubt what I have learnt so far. For example, let us decompose matrix A into L and U (LU factorization) $A=\left( \begin{array}{cccc} 5 & 4 & 1 &...
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1answer
18 views

Is square root of a matrix necessarily Cholesky decomposition? Or other decomposition satisfies?

If $a$ is a scalar, $a = \sqrt{a}\sqrt{a}$, then if A is a matrix, $A=\sqrt{A}\sqrt{A}$, now I think the square root operator is applied to the matrix instead of a scalar. Is square root of a matrix ...
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10 views

Sufficient condition for Unitary equivalence

Let $A = (a_{ij} )$ and $B = (b_{ij} )$ be similar, and $\sum ^n _{i,j=1} |a_{ij} |^2 = \sum ^n _{i,j=1} |b_{ij} |^2 $. Then A and B are unitary equivalent. I have to prove or disprove this ...
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A matrix is similar to its transpose without jordan form

Let $A \in M_{n×n}\mathbb{(C)}$ be invertible. Then $(A^∗)^{−1} = (A^{−1})^∗$ Let B be a nonsingular matrix such that $A = B^{−1}B^∗$ . Show that $A^{−1}$ is similar to $A^∗$. ($A^*$ is ...
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29 views

Eigenvector relations between matrices whose Gramian Matrices are the same

I have two symmetric real matrices I am interested in, $A_{n\times n}$ and $B_{m\times m}$. If I do the following operations: $$EAE^{T}=FBF^{T}$$ where $E$ and $F$ are of dimensions $p\times n$ and $...
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1answer
32 views

Decomposition of a diagonal matrix

I want to decompose a diagonal matrix $\Lambda \in R^{n \times n}$ such that $$ \Lambda \approx A\Sigma A^T $$ where $\Sigma \in R^{k \times k}$ is a diagonal matrix and $A \in R^{n \times k}$ is a ...
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9 views

Cholesky and $P^T LDL^T P$ decompositions not corresponding

I am using the Eigen library to factorize a matrix $A$ both using the Cholesky and the $LDL^T$ factorization. Basically, I correctly compute a matrix $C$ and check that $$A=CC^T,$$ then I correctly ...
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1answer
26 views

Singular vectors of a symmetric block secondary diagonal matrix

Given $A \in \mathbb{R}^{n \times m}$, consider the symmetric matrix $M = \begin{pmatrix} 0 & A \\ A^{t} & 0 \end{pmatrix} \in \mathbb{R}^{(n+m) \times (n+m)}$. Show that a simple ...
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Cholesky Decomposition and Equal Diagonals

If we know that a matrix $A$ has a unique Cholesky decomposition $L$, i.e. $A = LL^{T}$, and all the diagonal elements of $L$ are equal, can we say anything about the original matrix? I’m ...
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25 views

Change of largest eigenvalue by change of single element of nonnegative square matrix

I have matrices like A and B. Rows $2-(n-1)$ are stochastic, row $n$ is sub stochastic and the first row differs always in the first entry. I am interested in the largest eigenvalue ($\lambda_1$) of ...
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1answer
31 views

Find the inverse of a matrix by using the inverse of another one, which only differs in one row/column

In this question, one can see that it is possible to use the inverse $A^{-1}$ of $A = \begin{bmatrix} a & a_i \\ a_j & B \end{bmatrix}$ to calculate $B^{-1}$ with a simple $2\times2$ inversion....
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Rectangular matrices

Let $A$ is a $m \times n$ matrix factorized into $A = UV'$ where $U$ is $m \times k$ and $V$ is $n \times k$. I define the group of matrices $(UR,VR^{-1})$ where $R \in O(k)$ i.e. orthogonal group of $...
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Time complexity of least squares curve optimization using QR decomposition with Householder method.

Given a set of $m$ pairs of points, $<x_k, y_k>$, and a curve $y=ax^4 + bx^2 +c $ use the least squares method with QR decomposition and the Householder algorithm to approximate the curve's ...
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21 views

Calculate matrix in equation with constraints on values

Let's say I have the matrices $A = B \cdot C$ The matrices $A$ and $C$ are given, they contain real values $\in[0,1]$. I want to calculate the matrix $B$, which values can only be binary, $\in\{0$, $...
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2answers
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Why is the eigenvector matrix $S$ invertible?

I am currently learning Linear Algebra and I reached the topic of matrix diagonalization. What I understood that we do with matrix diagonalization is the following (please correct me if I am wrong): ...
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1answer
25 views

F any unitary matrix $U$ and integer $k>0$, there exists a polynomial $p(t)$, such that the matrix $B=p(U)$ satisfies the equation $B^k=U.$

Prove that for any unitary matrix $U$ and integer $k>0$, there exists a polynomial $p(t)$, such that the matrix $B=p(U)$ satisfies the equation $B^k=U.$ I am learning spectral theorem and SVD. I ...
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29 views

Updating Cholesky decomposition when deleting one row and one and column.

I found the answer of updating Updating Cholesky decomposition when deleting one row and one and column on Cholesky decomposition when deleting one row and one and column. Is there any generalisation ...
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Factorization of a complex-valued positive matrix into product of Hankel matrices.

I am dealing with a sort of "variance-covariance" matrix that surges from the multiplication of the Hankel matrix of a complex-valued time series and it's conjugate transpose. $$R_X=\mathcal{H}\...
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2answers
40 views

How to extract rotation matrix and scale vector from a 3D affine transformation?

For the affine transformation: $$ \begin{bmatrix} a & b & c\\ e & f & g\\ i & j & k\\ \end{bmatrix} $$ how do I extract the rotation and scale parts? According to ...
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what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works ...
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1answer
53 views

How to calculate $R=M(M^tM)^{-1/2}$ when one of the eigenvalues of $M^tM$ is $0$?

Suppose that I want to calculate the closest rotation in $SO(n)$ to an $n \times n$ matrix $M$. It can be shown by geometric arguments that $R = M(M^tM)^{-1/2}$. However, this requires that we can ...
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Generic “wavelet lifting” style matrix factorizations?

As context I will present the wavelet lifting scheme. In matrix terms it is a factorization of convolution that splits into two parts: Predict ($P$) step. Update ($U$) step. $$S=\begin{bmatrix}I&...
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15 views

Analogue of lograithm for matrix and vector multiplication?

Suppose $M$ is a $k\times k$ matrix and $v$ is a $k\times 1$ vector. What I want to know is whether there are functions $f$,$g$ and $h$ with $f: \mathbb{R}^{k\times k} \to \mathbb{R}^{k}$ and $g: \...
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1answer
32 views

Find first element of inverse matrix knowing Cholesky decomposition.

Given Cholesky decomposition of matrix A = LDL$^{T}$ = $A^{T}$ provide a possibly most efficient method to calculate upper left element of $A^{-1}$. I was thinking that this could be solved by using ...
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1answer
28 views

How is full pivoting more stable than partial pivoting

I know the difference between partial and complete, and I've read that complete is more "stable" / offers more "stability". In what sense is it more "stable"?
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3answers
40 views

Trying to find QR decomposition of a matrix.

Let $A=\begin{pmatrix}2&1\\2&0\\1&1\end{pmatrix}$ find $Q$ and $R$ such that $A=QR$. First I tried to find $Im(A)=\{(a,b,c)\in\mathbb{R}^3|Ax=(a,b,c)compatible\}.$ so then solving a ...
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2answers
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LU decomposition of SPD matrix without partial pivoting?

I get why diagonal dominant matrices do not need partial pivoting before Gaussian elimination can be applied in order to gain a LU decomposition, but why is this also the case for SPD matrices in ...
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1answer
46 views

Matrix Lower-Rank Factorization for L'L = A?

Assume we have Real, Symmetric, and PSD matrix $\textbf{A} \in \mathbb{R}^{n \times n}$, and $\textbf{A}$ has rank $r, \; r < n$. Then, $\textbf{A}$ will have the factorization of, $\textbf{A} = \...
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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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Schur factorization from spectral decomposition

Let $A \in \mathbb C^{d \times d}$. The Schur factorization of $A$ is the decomposition $A = U T U^\ast$, where $T \in \mathbb C^{d \times d}$ is an upper triangular matrix and $U \in \mathbb C^{d \...
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number of operations necessary to solve a matrix with different direct algorithms

I have the 4 following matrices that I have to solve using direct algorithms $$ \textbf{A} = \begin{bmatrix} 2 & -1 & & & \\ -1 & 2 & -1 & & \\ & -1 & 2 & -...
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1answer
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Condition number bound using block LU factorization

Let $A \in \mathbb{C}^{n \times n}$ be invertible and $P$ be a permutation matrix such that $$PA = \begin{pmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} $$ Where $A_{11} \in \...
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32 views

QR decomposition of linearly dependent vectors

Suppose $\vec{q_1},\vec{q_2},\vec{q_3}$ are linearly independent column vectors with dimensionality 4. $ X=[\vec{q_1}, \vec{q_2},(3\vec{q_1}-2\vec{q_2}),\vec{q_3}] $ then $X$ is a $4\times4$ ...
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24 views

Obtain unknown matrix from eigendecomposition.

I have doubt with the mathematical tools used in a problem of Signal Analysis: I have a complex observable series $Y(t)$ which is the result of summing two complex r.v $X(t)$ (unobservable) and a $\...
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1answer
15 views

$LDL^\top$ for symmetric positive semidefinite matrices that are not positive definite

I have a symmetric positive semidefinite matrix (which is not positive definite) with integer entries and I know that I have an $LDL^\top$ decomposition for it (well mainly because Maple gives me one)....
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1answer
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Singular Value Decomposition of this matrix with a zero singular value

What is the singular value decomposition of $$ \left[ \begin{matrix} 2 & -6\\ 1 & -3 \\ 0 & 0 \\ \end{matrix}\right] $$ Singular value decomposition calculators online ...
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Directly prove strictly column diagonally dominant matrix is nonsingular using matrix norms

It is the case that, if a matrix $A \in \mathbb{C}^{n \times n}$ is strictly column diagonally dominant, meaning $$|a_{jj}|> \sum_{i \neq j}^{n}|a_{ij}|, \quad 1 \leq j \leq n $$ then it is ...
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1answer
64 views

Matrix equation of the form $C A C^\intercal = D$

Consider the following square matrix \begin{align} A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\ 0 & d & -d+1 & -\frac12 & 0 & ...
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36 views

Find nth power of symmetric matrix of size m*m where m<=1000 and n can be any positive integer?

i have to find nth power of matrix which is symmetric about main diagonal and all elements of main diagonals are zero. For example sample matrix $\left( \begin{array}{cc} 0 & 1 & 0 & 0\\ 1 ...
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Fundamental matrix of a markov process with binomial transition probabilities

I'm studying a Markov process with $n$ transient states and one absorbing state (indexed $1, \ldots, n+1$), with transition probabilities given by the $n+1\times n+1$ matrix $P$ where the probability ...
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What can I say about $\mathbf{X}$ if $\mathbf{y}^{\mathrm{T}} \mathbf{X} \mathbf{y} = \mathbf{z}^{\mathrm{T}} \mathbf{X} \mathbf{z}$?

Consider the $N$-dimensional positive semidefinite matrix $\mathbf{X}$ and two $N$-dimensional vectors $\mathbf{y}$ and $\mathbf{z}$ with $\|\mathbf{y} \|^{2} = \| \mathbf{z} \|^{2}$. If I know that $\...
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1answer
33 views

If $A$ is positive semidefinite, then $A=USU^T$?

If A is a positive semidefinite matrix, then it has a singular value decomposition $A=USV^T$ with $U=V$. My textbook states this as fact, but I cannot seem to prove it. Additionally, $S$ must have the ...
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1answer
62 views

Write the derivative of the lower-triangular matrix $L(t)$ in terms of $L(t)$, $L^{-1}(t)$ and $\frac{d}{dt}A(t)$, where $A(t)=L(t)L^T(t)$

Let $A(t)$ be a symmetric positive definite matrix, thus by Cholesky decomposition, we have $A(t)=L(t)L^T(t)$ where $L(t)$ is lower triangular. Suppose $A(t)$ is differentiable. I want to write $\...
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1answer
109 views

Matrix filled with 0's and 1's

A matrix of size $n \times m$ filled with $0$'s and $1$'s and this matrix is considered lucky if no two adjacent cell are same i.e.,( no adjacent two 0's or 1's together ) so, we can invert some ...
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1answer
22 views

Summation of polynomial matrix multiplication in terms of vector outer product

Consider the following summation $$ \sum_{i=1}^{T-1}C(A^i-A^{i-1})Bx_{t-i} $$ where $A$ is a $d \times d$ diagonal matrix, i.e. $A=\text{diag}(\alpha_1,\cdots,\alpha_d)$, $C$ is an $m \times d$, $B$ ...
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16 views

Vandermonde decomposition of Toeplitz matrix

Since every Hermitian Toeplitz $A \in C^{n \times n}$ with $rank(A)=K$ can be decomposed as $$ A = VDV^T $$ where $V \in C^{n \times K}$ is a Vandermonde matrix, and $D \in R^{K \times K}$ is a ...
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2answers
98 views

LU factorization of a nonsingular matrix exists if and only if all leading principal submatrices are nonsingular.

I'm struggling to prove this theorem. I can prove that if the $LU$ factorization exists, then the leading principal submatrices are nonsingular. To do that, I can show that the determinant of every ...
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1answer
23 views

Column Orderings in any QR/Matrix Factorization Method

I am trying to understand if the ordering of columns matters in QR decompsoition. In general it seems that column ordering won't matter. I guess for SVD or any matrix factorization the way columns ...
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0answers
15 views

Truncated SVD matrix approximation to block dense matrix

Given a symmetric p.s.d matrix $A$ where every element is greater than 0, and $A$ has a block diagonal structure such as: $$ A = \begin{matrix}\begin{pmatrix}A_{11}, A_{12} \\ A_{21}, A_{22}\end{...