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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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1answer
14 views

Is $A+D$ an irreducible matrix?

Suppose that $A$ is an irreducible matrix and all its entries are non-negative. Let $D$ be a diagonal matrix whose all entries are positive. Is $A+D$ an irreducible matrix? Since $A$ is an ...
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0answers
9 views

Equivalence of semidefinite decomposition?

If we have an $n \times n$ positive semidefinite matrix $A$ and we have two decompositions such that $A = B B^T = C C^T$ for some $n \times n$ matrices $B$ and $C$. Is it true that $B$ and $C$ are ...
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0answers
30 views

Limit of Matrix power equals $\pi$ [on hold]

Denote matrix \begin{equation} A= \begin{pmatrix} 1-t&2-t\\-t&1-t \end{pmatrix} \qquad (0<t<1) \end{equation} and vector $u_n\equiv(u_{1n},u_{2n})^\top=A^n v$, where $v=(1,1)^\top$. $N(t)...
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0answers
23 views

Band matrix conjugation relation between orthogonal matrices

Suppose I have two orthogonal matrices: $C_1,C_2$ of some dimension $n$. It is given that $C_2 = R_1*C_1*R_2$ where R1,R2 are orthogonal matrices as well. How can I solve for $R_1$ and $R_2$ ...
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1answer
23 views

A matrix equation involving eigendecomposition

Let $p<n$ and -$H\in\mathbb{R}^{n\times n}$ be symmetric with eigendecompoistion being $H=U\Lambda U^{\text{T}}$, -$A\in\mathbb{R}^{n\times p}$, -$D\in\mathbb{R}^{p\times p}$ be a diagonal ...
2
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1answer
69 views

Householder transformations to upper triangular form

Let $A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$. How to transform this matrix with Householder transformations to an upper ...
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0answers
11 views

Condition for positive definiteness of matrix difference

Suppose I have two matrices $X \in \mathbb{R}^{n \times q}$ and $Y \in \mathbb{R}^{n \times q}$. Suppose that both $X$ and $X - Y$ have rank $q$ but $Y \neq 0$. I want to determine if there are any ...
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1answer
36 views

Determinant of a large symmetric block matrix

Consider a given matrix $Q \in \text{Mat}_N(\mathbb{R})$, which is invertible, and $n \geq 1$. I am looking for the determinant of the symmetric block matrix $I_n(Q)$ of total size $nN \times nN$: $$...
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1answer
31 views

Is there a closed-form formula for the derivative of the orthogonal polar factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\newcommand{\SO}{\operatorname{SO}_n}$ $\...
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4 views

Is this the correct iterative version of the multiplicative update rule for matrix factorization?

So we have $A_{m\times n}\approx P_{m\times k}Q_{k\times n}.$ Using the multiplicative update rule due to Lee we, in general, write that: $$P\leftarrow P \circ \frac{AQ^T}{PQQ^T}$$ and $$Q\leftarrow ...
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23 views

Existence of a QR factorization

Let $A \in \mathbb{R^{n \times n}}$ be an nonsingular matrix and $LL^T$ the Cholesky decomposition of $A^TA$. How to show that it exists a QR factorization with $Q=A(L^T)^{-1}$? I tried this: $A^TA=...
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12 views

How can I find the Schur decomposition from QR factorization?

Let's say that I got a matrix $A$ and I find the $Q$ and $R$ matrix from QR-factorization. $$A=QR$$ Now I want to find the Schur decomposition. $$A=QUQ^{-1}$$ How can I do that? Is it the same $Q$...
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16 views

How to prove that polar decomposition matrix is symmetric

Given polar decomposition of an invertible matrix F=RU where R is an orthogonal matrix. How to prove that U is symmetric?
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20 views

Inequality with a QR decomposition [duplicate]

Let $A \in \mathbb{R^{n \times n}}$ be an invertible matrix. How to prove $|$det$(A)$|$\leq \prod_{j=1}^{n}{(\sum_{i=1}^{n}{|A_{ij}|^2})^{\frac{1}{2}}}$ with a QR decomposition? I tried to use: ...
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2answers
52 views

Proof of Jordan-Chevally-Decomposition

Let A be a square matrix over $\mathbb{C}$, prove there are matrices $D$ and $N$ such that $A = D + N$ such that $D$ is diagonalizable, $N$ is Nilpotent and $DN = ND$. I can see that any ...
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1answer
10 views

Is the transformation matrix of an upper triangular matrix to its Jordan normal form always triangular?

Assume that $A$ is an upper triangular matrix. In the case where $A$ is 2x2, I've checked that a transformation matrix $P$ such that $J = P^{-1}AP$, with $J$ Jordan normal form of $A$, is always upper ...
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2answers
106 views

How to prove this very interesting matrix identity?

This is a very interesting identity but I don't know how to prove this, note that $$A_1,\ldots,A_J,B \in \mathbb{R}^{n \times n}$$ and $m$ is the number of block diagonal in $\mathbf{A}$ ,so consider ...
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0answers
9 views

Recursive QR-factorisation for N4SID - What does this equation mean?

I was reading a paper about recursive subspace identification, where they are using N4SID-algorithm with some extantion for the recursive method. http://www.iaescore.com/journals/index.php/IJEECS/...
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34 views

Sufficient condition on matrix to tridiagonalize

consider a possibly infinite quadratic matrix $A$. Based on this post: Decomposition into a tridiagonal matrix I am wondering whether there are sufficient conditions aside from $A$ being symmetric ...
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2answers
34 views

the inverse of a sum of two symmetric for schur completion?

I have a up-triangulate Jacobi matrix J which can be blocked like : $J = \begin{bmatrix}A & B\\ 0 & C\end{bmatrix} $ both A and C are up-triangulate, we can get Hessian matrix H by: $H = J'...
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1answer
37 views

Is this pseudo-Cartan decomposition of $SO(n)$ valid?

I'm a graduate student in a field of science where we frequently need to optimize a matrix in $U(n)$ or $SO(n)$ (henceforth $SO(n)$ for concreteness) to get an "optimal" orthonormal basis before doing ...
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0answers
22 views

Perturbation theory of the eigenvalues about a symmetric matrix: Reality of Eigenvalues

Let $A$ be an $n\times n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $\hat A := A+E$ are all ...
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1answer
28 views

Real Matrices with Real Eigenvalue pre- and Post multiplied by a Diagonal Matrix

Suppose all the eigenvalues of $A\in \mathbb{R}^{n\times n}$ (not necessarily symmetric) are real. Let $D\in \mathbb{R}^{n\times n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $...
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2answers
91 views

Invert a $4 \times 4$ matrix with a given structure?

When tryig to fit $f(x,y) = a+bx+cy+dxy$ to the values of four points, we will have to invert following matrix. (Let us assume that $x_i,y_i$ are chosen suitably such that it is regular.) We could ...
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1answer
53 views

Why SVD is not unique but the Moore-Penrose pseudo inverse is unique?

I feel confused about the uniqueness of the Moore-Penrose inverse generated from SVD. For any matrix $A$, if $X$ satisfied $$AXA=A, XAX=X, (AX)^\mathrm{T}=AX, (XA)^\mathrm{T}=XA $$then $X$ is called ...
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0answers
11 views

LDL factorization of symmetric indefinite banded matrix

I have a symmetric indefinite matrix $A$ which is banded, and I want to compute the $LDL^T$ factorization, however there do not appear to exist codes to do this in standard libraries (such as LAPACK, ...
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0answers
15 views

QR factorization $\;AA^T x = b$

I would like to solve the following system with QR factorization of $A^T$. $ A A^T x = b$ with $ A \in R^{m \times n}, \;x \in R^m\;$ and $b \in R^m$. In my case is $n>m$. I know how to do a ...
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0answers
16 views

How does the matrix multiplication change the smallest singular value?

Assume the matrix $X \in R^{M\times N}$ has rank $r$. Suppose we have an arbitrary semi-unitary matrix $W \in R^{M\times r}$ such that $W^T W = I$. In general, the matrix $Y=W^TX \in R^{r\times N}$ ...
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38 views

Inequality of matrix infty-norm: Does $\|(I - L)^{-1}U\|_{\infty} \leq \|H\|_{\infty} < 1?$

If $H\in \mathbb{R}^{n\times n}$ is a matrix with $\|H\|_{\infty} < 1$ and $H = L + U$, where $L$ is the strict lower triangular part of $H$ and $U$ is the upper triangular part of $H$, can we ...
7
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1answer
113 views

Solving the matrix equation $X^tA+A^tX=0$ for $X$ in terms of $A$

Suppose that I know $A$. And all matrices in the equation are square matrices. I want to solve for $X$ given that $$X^tA + A^tX = 0$$ I'm not really good at matrix calculus. Is it possible to solve ...
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0answers
30 views

Proof of existence of LU-decomposition

I have a question concerning an existence proof of the $LU$-decomposition. The proof is as follows: If $E_{ij}$ denotes the matrix with $1$ at row $i$, column $j$ and zeros elsewhere then I let $P$ ...
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1answer
45 views

Find diagonal matrix $D$ such that $A D$ is Hurwitz

Let $A \in \mathbb{R}^{m \times m}$. Give necessary and/or sufficient conditions for the existence of a matrix $D \in \mathbb{R}^{m \times m}$ such that all eigenvalues of $AD$ have negative real part ...
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1answer
30 views

Decomposition of a symmetric matrix $xx^T$ into a rank one and residual matrix?

Suppose we have $M=xx^T$ where $x$ is a random vector in $\mathbb{R}^n$. Also, we know that $x=q+e$ where $q$ is distributed according to $D$, i.e., $q \sim D$ and $e$ is a bounded vector. Therefore, $...
4
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1answer
125 views

Is the orthogonal polar factor the unique submersion satisfying an orthogonality relation?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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1answer
24 views

Is the orthogonal polar factor the unique retraction $\operatorname{GL}_n^+ \to \operatorname{SO}_n$?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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24 views

Relation between SVD and POD

I understand that POD is about choosing an optimal base, and i have found this Eckart Young theorem And also have encountered on a book that given an matrix $A$ its projection on POD modes is given ...
2
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0answers
44 views

Is there a closed-form formula for the derivative of the positive factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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1answer
43 views

Is it true that $A^{T}P+PA>0$ for an unstable matrix $A$ and a positive definite $P$?

For a positive definite matrix $P$ and a matrix $A$ with all positive eigenvalues, how to guarantee that the matrix $Q=A^{T}P+PA$ is positive definite? I know if $A$ is a stable matrix (i.e. all the ...
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0answers
19 views

Is there a problem with the “low-rank matrix approximation” to have low rank?

I know that "rank" is the number of independent rows in the matrix and I know that the resulting matrix of the "low-rank matrix approximation" algorithm has lower rank than the original matrix. 1- But ...
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2answers
25 views

Why take $\sqrt {A^ \dagger A}$ in the polar decomposition of matrix A?

In the Polar Decomposition section in Nielsen and Chuang (page 78 in the 2002 edition), there is a claim that any matrix $A$ will have a decomposition $UJ$ where $J$ is positive and is equal to $\sqrt{...
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0answers
11 views

Prove that all normal matrices are semi-simple using Schur's decomposition

Is there any elegant proof that shows that all normal matrices are semi-simple that comes from Schur's decomposition or its corrolaries? There is a proof that normal matrices are unitary diagonizable ...
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1answer
24 views

Multiplications that preserve singular values

What is the characterization of matrices $B$(not necessarily squared) such that $BA$ has the same largest singular value as $A$? How about when $BA$ preserves the same $k$ largest singular values of $...
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0answers
32 views

SVD of a specific matrix and Singular values behaviour

I have a rectangular matrix $A \in \mathcal{M}_{l,n} (\mathbb{C})$, $l>n$ which has this property : $$A=\left[\begin{matrix} M_1 & i M_2 \\ M_2 & -i M_1 \end{matrix}\right]$$ Where $M_i$ ...
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0answers
5 views

matrix decomposition (not quite non-negative matrix decomposition)

I'm interested in finding a good approximation of the mxn matrix $Y$ as $Y\approx XB$ where the mxn matrix $X$ has only non-negative elements and $B$ is an invertable nxn matrix. Are there algorithms ...
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1answer
29 views

If the off-diagonal entries of a positive-definite symmetric matrix $A$ are $\leq 0$, then $A^{-1}$ has positive entries

Let $A \in \operatorname{GL}_n(\mathbb R)$ be a symmetric matrix which is positive definite, i.e. $A = Q^tQ$ for some invertible matrix $Q$. Suppose that the off diagonal entries of $A$ are $\leq 0$. ...
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0answers
17 views

Finding the Upper Bound of the difference between the Inverse of the 2 matrix

Given that $ K = A^{-1} - B^{-1} + A^{-1} - A^{-1}BA^{-1}$, we need to find the upper bound of $K$ where matrix $A = C + I\rho$ and $ B = C_{x} + I\rho $ has dimension $n\times n$, $ C = RDR^{T}$ ...
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0answers
49 views

Subgroup generated by union of two maximal compact subgroups of $GL_2(\mathbb{Q}_p)$

Let us denote by $G:= GL_2(\mathbb{Q}_p), G_0:= GL_2(\mathbb{Z}_p), g:= \begin{bmatrix}0 & 1\\p& 0\end{bmatrix}$ and by $G_1:= g G_0 g^{-1}$. I want to know if we have a good description of ...
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2answers
52 views

How to inverse a block diagonal matrix?

Given a matrix $$x = \begin{bmatrix} 40 & 0 & 0 & 0\\ 0 & 80 & 100 & 0 \\ 0 & 40 & 120 & 0 \\ 0 & 0 & 0 & 60\end{bmatrix}$$ How to find the inverse of ...
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0answers
20 views

Norm of least squares residual

Let $A=QR$ be the full $QR$ factorization of $A\in\mathbb{C}^{m\times n}$, $m\ge n$, where $$Q=\begin{bmatrix} \hat{Q}_1 & \hat{Q}_2 \end{bmatrix}, R=\begin{bmatrix} \hat{R}_1\\ 0 \end{bmatrix},$...
1
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1answer
22 views

Finding the polar decomposition of a $2\times2$ matrix.

Specifically, the question is as follows: Let $B$ be the standard basis of $\mathbb{R}^2$. Let $T\in\mathcal{L}(\mathbb{R}^2)$ be such that $$\mathcal{M}(T,B)=\begin{bmatrix}2&3\\0&2\end{...