Questions tagged [singular-values]

This tag is for questions relating to 'Singular Value'. The term “singular value” relates to the distance between a matrix and the set of singular matrices

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24 views

For non compact operators, is countability of singular values equivalent to countability of eigenvalues?

Generally speaking, compact operators have countable eigenvalues and countable s-values (or singular values). What about the reverse? If I know that a (non-compact) operator has countable singular ...
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1answer
14 views

A and B have identical singular-values and identical eigenvalues, are they unitary similar?

I've come across this question while taking a matrix analysis course. Given: $A,B \in\mathbb{C}^{3 \times 3}$ with $\lambda_1(A) = \lambda_1(B) ,\,\lambda_2(A) = \lambda_2(B),\,\lambda_3(A) = \...
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37 views

(SVD) why does the first right singular vector minimise the sum of squared distances to a projection?

for days now i've been searching the whole internet to find an answer on this question but I didn't find anything. Can anyone please help me on that or give me a hint? I need to prove that the first ...
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38 views

Is the truncated Frobenius norm convex or not?

Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated Frobenius norm of a matrix $X$ is defined by $$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$ where ...
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1answer
26 views

What is a sufficient condition for the singular values of a square matrix to be the modulus of its eigenvalues?

My lecturer said " when the matrix M is square and it has complex-conjugate eigenvalues(I mean conjugate pairs of eigenvalues for any non-real eigenvalue), the singular values of M are the ...
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19 views

Limitation through singular value of a matrix

Question: Given a matrix $X \in \mathbb{R}^{m \times n}$. Let $(X^k)_k \in \mathbb{R}^{m \times n}$ is a sequence converge to $X$. Denote $\sigma_i(X)$ as singular values of $X$. Prove that $$\lim_{k \...
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1answer
31 views

Smallest Euclidean norm after matrix multiplication

If you multiply a vector $x$ with a matrix $A$, where $x$ has unit Euclidean norm, is there any direction in which $x$ can point, which results in a smaller resulting magnitude than the smallest ...
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1answer
41 views

Matrix that maximizes that trace problem [duplicate]

I am trying to prove that the optimal solution for the following problem $$\begin{array}{ll} \underset{X \in \Bbb R^{n \times m}}{\text{maximize}} & \mbox{Tr} (AX^T)\\ \text{subject to} & \|X\|...
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1answer
65 views

Why are the singular values non-negative?

I am working with SVD and trying to understand it. Through some thinking and help here I have a better understanding of it, but I would like some feedback. Thought process: $AA^T$ and $A^TA$ share ...
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25 views

Find all forms of the matrix whose right singular vector is $\frac{1}{\sqrt{n}}(1,\cdots,1)^{\top}$ corresponding to the largest singular value

Find all forms of the matrix $A \in \mathbb{R}^{m \times n}$ which satisfies $A \mathbf{v}^{0}=kA \mathbf{v}$, where $\mathbf{v}=\frac{1}{\sqrt{n}}(1,\cdots,1)^{\top}$ is a right singular vector ...
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Let $A\in\mathbb C^{n\times n}$ be normal with $n$ eigenvalues and singular values. Show that $\sigma_i(A) = |\lambda_i(A)|$ for $i\in n$.

For some unitary $U$, we have $$U^HAU = D = \text{diag}(\lambda_1,\dots, \lambda_n)$$ Then $$ DD^H = U^HAUU^HA^HU = U^HAA^HU $$ how do we then conclude that $\sigma_i^2(A) = \lambda_i(A)\overline{\...
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14 views

Non trivial lower bound for $\sup_P \|Px\|$ where $x$ is fixed, and the supremum if over all full-rank projection matrices $P \in R^{k \times n}$

Let $1 \le k \le n$ be integers and let $G_{n,k}$ be the grassmannian of $k$-dimensional subspaces of $\mathbb R^n$. For any $U \in G_{n,k}$, let $P_U$ be the orthogonal projection onto from $\mathbb ...
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1answer
30 views

Upper-bound for nuclear norm of $A \circ (v \otimes v)$ in terms of operator norm (or nuclear norm) of matrix $A$ and $L_\infty$-norm of vector $v$.

Let $A \in \mathbb R^{n \times }$ be a psd matrix such that $\|A\|_{op} \le r_1$ and $\|A\|_{*} \le r_2$. Let $v \in \mathbb R^n$ such that $\|v\|_\infty \le r_3$. Let $B:=A \circ V$ be the Hadamard ...
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22 views

Upper bound on condition number of real symmetric positive definite matrix $A^TA$ with bounded entries

Hi, I am looking for an upper bound of the condition number of a real symmetric matrix $A^TA$ with bounded entries. The matrix of interest is $A^TA$, where $A \in \mathbb{R}^{n \times p}$, $n>p$ is ...
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6 views

Is unit singular values equivalent to inner product preservation for non-square matrices?

Let $X$ be an $n×m$ matrix ($m>n$) with singular values all equal to 1. Then $X$ preserves inner products. That is, $x^TX^TXx=x^Tx$. What about the opposite direction. If it preserves inner ...
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14 views

Is there a name for matrices with singular values all equal to $1$?

I at first thought these are just the orthogonal matrices, since the SVD of $X$ is $U \Sigma V^T$ and if $\Sigma$ is the identity matrix then this implies X is orthogonal. However, singular values ...
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1answer
41 views

Special lower bound on minimum singular value

We know that any square matrix $M$ could be written in the form $H_1 + iH_2$ for some Hermitian operators $H_1$ and $H_2$. Denote the smallest singular value of a matrix by $\sigma_m(.)$. Can we lower ...
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35 views

How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?

Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...
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1answer
22 views

the minimum singular value

For any a matrix $X$, based on the Cauchy Interlacing Inequality below, I guess that \begin{equation} \sigma_{\min}(X) \leq \sigma_{\min}(Y), \end{equation} where $Y$ is any a sub-matrix of $X$. I don’...
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1answer
42 views

$\Lambda^k(A)$ is a trace class operator

Let $A$ be a trace class operator. I am trying to understand the proof of ($\|\cdot\|_1$ means the trace class norm) $$\|\Lambda^k(A)\|_1\leq \frac{\|A\|_1^k}{k!}$$ I have a proof but I think there ...
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19 views

Interpolation spaces defined by singular value decomposition

Let $ X $ and $Y$ be Hilbert space, $A:X \to Y $ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is, $$ Av_n = \sigma_n u_n \\ A^* u_n = \sigma_n v_n $$ Since $\...
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77 views

Singular value decomposition and rank with min max theorem for singular value

Let $A=U\Sigma V^T$ be singular value decomposition of $A\in \mathbb{R}^{m \times n}$, where U, V are orthonormal, $\Sigma=diag(\sigma _1,...,\sigma_r,0,...,0)$ with $\sigma_1 \geq...\geq \sigma_r \...
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42 views

How to get the singular value of a 2-by-2 matrix?

I am going to get the principal angles between 2 subspaces in $\mathbb{R^4}$ according to this hint. The 2 matrices involved are $A = \begin{bmatrix} a & -b \\ b & a \\ c & -d \\ d & c ...
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1answer
19 views

How to prove this inequality consisting of the minimum singular value? [closed]

How does one proceed in solving this inequality? X is of full column rank $$\left\|\left(\boldsymbol{X}^{T} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{T} \boldsymbol{v}\right\|_{2} \leq \frac{1}{\...
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52 views

Upper bound on singular values of Kahan matrix

I have been going through the paper by Gu, Ming, and Stanley C. Eisenstat. "Efficient algorithms for computing a strong rank-revealing QR factorization." SIAM Journal on Scientific Computing ...
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8 views

Prove: $n^{-1} tr(DG^{-2}-D_0G_0^{-2}) \to 0$ if $D_0,G_0\in R^{n \times n}$ have bounded spectra, and $\|D-D_0\|,\|G - G_0\| \to 0$ as $n \to \infty$

Let $n > 0$ be a large integer and let $D,D_0,G,G_0$ be $n \times n$ psd matrices such that $\|D-D_0\|_{op} = o_n(1)$ and $\|G-G_0\|_{op} = o_n(1)$. $\|D_0\|_{op},\|G_0\|_{op},\|G_0^{-1}\|_{op} = ...
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23 views

how to give an upper bound on the minimum singular value of the derivative of Vandermonde matrix

Now I have a matrix $\begin{equation}\label{key} R = \begin{bmatrix} 0 & 0 & \cdots & 0 & \\ 1 & 1 & \cdots & 1 \\ 2a_1 & 2a_2 & \cdots & 2a_n \\ \vdots & \...
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1answer
26 views

Is possible set up a homeomorphism over a singular isolate point?

$\textbf{My question:}$ Let $f:U\rightarrow R^n \in C^1$ with $U\subset R^n$ open. If $a$ is a singular point of $f$ such that $a\in B(a,\epsilon )$ and every point in $B(a,\epsilon )-\left\lbrace a \...
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1answer
33 views

Singular Vector Decomposition and PCA interpretation

The covariance matrix of any data X -> N * D (N samples and D dimension ) would be Cov(X) = E[(X - E[X])(X - E[X])T]. Let's Assume (X - E[X])=Y. Thus Cov(X) = E[YYT]. Now from SVD we know that U ...
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1answer
60 views

Showing the Relationship between singular values for two matrices

Let $A\in \mathbb{C}^{m\times n}$ with $m> n,$ $z \in \mathbb{C}^{m\times 1}, \ B = [{A \ z}].$ Where $\sigma_i$ are the singular values. Show: (a) $\sigma_1 (B)\ge \sigma_1 (A)$ and (b) $\sigma_{...
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30 views

Eigenvalues of Differences of Hermitian Matrices $A, B$ When Trace $Tr(A - B) = 0$

Let's say we have two Hermitian matrices $A$ and $B$. We know the eigenvalues of both $A$ and $B$. We also know that $A - B$ is traceless, meaning: $$ Tr(A - B) = 0. $$ A piece of additional ...
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75 views

Gradient of spectral norm of matrix

I am looking for a reference for the following result: Let $A$ be an $m \times n$ real matrix. Let $\sigma(A)$ be the spectral norm of $A$. If the largest singular value of $A$ is unique, then $$ \...
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93 views

On what condition on $L$, there exists unit vector $x$, such that $\Re(x^∗Lx)<0$ and $\Im(x^∗Lx)=0$?

Given $L\in\mathbb{C}^{n\times n}$. Question 1: On what condition on $L$, there exists unit vector $x\in\mathbb{C}^n$, such that $x^*L^*Lx=1$. Question 2: On what condition on $L$, there exists unit ...
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25 views

On the characterization of the least singular value

Let $A$ be an $m\times n$ real or complex matrix. Here it is shown that $$\sigma_{min}(A)=\min_{x\in F^n, \|x\|=1}\|Ax\| \quad \quad (1)$$ where $\sigma_{min}(A)$ denotes the "least" ...
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44 views

Show the relation between singular values of $A$ and $B^*AB$

Proposition: For any nonsingular $A\in\mathbb{C^{n\times n}}$ and nonsingular $B\in\mathbb{C^{n\times n}}$, the following holds $$\frac{\Re(x_i^*Ax_i)}{\sigma_i(A)}=\frac{\Re(y_j^*B^*ABy_j)}{\sigma_j(...
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52 views

Why is a left-singular vector in the kernel, and a right-singular vector in the cokernel?

Some $m\times n$ matrix $M$ has a singular value decomposition $M=U\Sigma V^H$, where $U,V$ are unitary and $\Sigma$ is a rectangular diagonal matrix with only real entries. $^H$ means conjugate ...
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354 views

When do Linear Transformations NOT preserve angles between vectors? Doesn't the SVD tell us all linear transformations preserve angles?

From searching on the internet, I learned only a subset of linear transformations preserve angles between vectors. But - Learning about the SVD - we can geometrically understand as breaking down some ...
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23 views

In singular value decomposition, is there a difference between starting with transpose(A) * A or A * transpose(A)?

When calculating the SVD for a matrix I go through these steps : 1/ Calculate tranpose(A)*A 2/ Find the eigenvalues for that matrix and deduce $\Sigma$ 3/ Find the eigenvectors and deduce $V$ 4/ ...
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1answer
204 views

Minimize $x^*(A+A^*)x$ such that $x^*A^*Ax=1$ and $x^*x=1$

Given $A\in\mathbb{C}^{n\times n}$, such that it has singular values larger than $1$ and smaller than $1$, \begin{array}{ll} \underset{x\in\mathbb{C^n}}{\text{minimize}} & x^*(A+A^*)x.\\ \text{...
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35 views

How can I prove a randomly generated matrix has distinct non-zero eigenvalues?

Consider the following $M \times M$ matrix $$\mathbf A = \sum_{k=1}^K a_k \mathbf h_k \mathbf h_k^H, \quad (M \geq K)$$ where $a_k$'s are real values and $\mathbf h_k$'s are $M \times 1$ randomly ...
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1answer
59 views

$\|K\|=\max_{j}s_{j}(K)$ for Compact Operator

Could anyone reference me to a proof of $$\|K\|=\max_{j}s_{j}(K)$$ $K$ being a compact operator and $s_{j}(K)$ being the singular values of the operator. I cannot find a proof in the literature.
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67 views

Expected norm of matrix product with random unit vector

Let $S_d = \{x \in \mathbb{R}^d : \|x\| = 1\}$ be the unit sphere in $\mathbb{R}^d$. Let $A \in \mathbb{R}^{n \times d}$. Note that \begin{align} \max_{x \in S_d} \|A x\| &= \max \sigma(A) &...
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1answer
115 views

Relationship between matrices whose singular values are the same

Motivation: I have two different matrices in $\mathbb{R}^{1000 \times 2048}$. $A_1$ is coming from an sparse optimization process whose objective is creating as much as zeros in $A_1$. In this sense, ...
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0answers
20 views

Find all $x\in\mathbb{C^n}$ such that $\Im ( x^*Ax )=0$ and $||x||=1$.

Given $A\in\mathbb{C}^{n\times n}$, find all $x\in\mathbb{C^n}$ such that $\Im ( x^*Ax )=0$ and $||x||=1$. My attempt: Any complex number can be decomposed to hermitian part and skew hermitian part: $...
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1answer
171 views

Find all $x\in\mathbb{C^n}$ such that $||Ax||_2=1$ and $||x||_2=1$

Given $A\in\mathbb{C}^{n\times n}$, find all $x\in\mathbb{C^n}$ such that $||Ax||_2=1$ and $||x||_2=1$. Lets do SVD: $A=U\Sigma V^*$, where $\Sigma=\mathrm{diag}\{\sigma_1,\ldots,\sigma_n\}$. We do ...
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1answer
39 views

Fast way to numerically compute smallest non-zero singular value for implicit matrix A

I have a matrix $A \in \mathbb{R}^{n \times m}$ implicitly defined (i.e. for which I can calculate the matrix-vector products $Ax$ and $A^T x$, but storing the entire matrix is prohibitive in terms of ...
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0answers
13 views

Singular values of a matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows. We wish to find a tight bound on the singular values. We know on the rows that they have (...
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1answer
37 views

Trying to understand why the largest singular value of $M_n^{-1}$ is $1.4286$ no matter what $n$ I choose

Let $n \ge 3$ be an odd number. Let $I_n$ be the $n$ dimensional identity matrix and let $A_n$ be the $n\times n$ matrix where every element is zero except the central element which is, say, $0.3$. ...
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0answers
41 views

How is SVD better than Gaussian elimination in finding the rank of a matrix?

In Linear Algebra and its Applications, Gilbert Strang, $4^{th}$ ed, one of the applications of SVD is mentioned as finding the effective rank of a matrix. The idea presented in the book is that the ...
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1answer
46 views

Matrix reconstruction puzzle

Say a reconstruction of matrix $A$ is $A'$ and it's defined as $ A' = P^TDPA $ where $P$ is an orthonormal matrix, $D$ is a diagonal binary (1 or 0) matrix. In a trivial case, when all diagonal ...

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