Fdtxjr

SOQL query causes internal Salesforce error

Sigmoid with a slope but no asymptotes?

Unable to disable Microsoft Store in domain environment

How were servants to the Kaiser of Imperial Germany treated and where may I find more information on them

How to get directions in deep space?

Why would five hundred and five be same as one?

Confusion over Hunter with Crossbow Expert and Giant Killer

Ways of geometrical multiplication

Why do Radio Buttons not fill the entire outer circle?

Echo with obfuscation

If the only attacker is removed from combat, is a creature still counted as having attacked this turn?

Can you identify this lizard-like creature I observed in the UK?

Storage of electrolytic capacitors - how long?

In One Punch Man, is King actually weak?

What the heck is gets(stdin) on site coderbyte?

Mimic lecturing on blackboard, facing audience

Should I warn a new PhD Student?

Proving an identity involving cross products and coplanar vectors

Isometric embedding of a genus g surface

Why does the Persian emissary display a string of crowned skulls?

How much do grades matter for a future academia position?

How do I prevent inappropriate ads from appearing in my game?

What's the name of the logical fallacy where a debater extends a statement far beyond the original statement to make it true?

Check if object is null and return null

Fix $v in mathbb R^n$, the probability matrix $A$ with normally distributed entries has an eigenbasis that $v$ projects nontrivially is $1$

trace class norms of random matricesCan't show that these matrices are diagonalizable.Show $A$ is diagonalizableGiven that $A$ is normal, show that there exists a matrix $B$ such that $B^k = A$, where $k in mathbb N$Hermitian Operators and the Spectral Theorem$A^topA = (A^topA)^2$ mistakeSquare root of operators over $mathbbC^n$Spectrum of a matrix with normally distributed entriesAre the sets $X: max_i textRelambda_i (B+AX) < 0$ and $X: rho(B+AX) < 1$ homeomorphic?Is the function $A mapsto sumlimits_j=0^infty langle A^j v, A^j v rangle$ differentiable everywhere?

2

$begingroup$

Suppose $v in mathbb R^n$ is a fixed nonzero vector and $A in M_n(mathbb R)$ is a random matrix where each entry is taken from a standard normal distribution over $mathbb R$. We know that the probability $A$ is diagonalizable is $1$. My question is: if we concern those matrices that are diagonalizable (thus with an eigenbasis), what is the probability that those with an eigenbasis and $v$ projects nontrivially on each eigenvector in the eigenbasis, i.e., the probability measure of the set
beginalign*
A in M_n(mathbb R): & texteach entry is generated from a standard normal,\
& textif $A=SLambda S^-1$ (spectral decomposition) then $Sv$ has no zero component.
endalign*

linear-algebra probability-theory random-matrices

share|cite|improve this question

edited Mar 14 at 5:34

MyCindy2012

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Unless $v=0$, the probability that $v$ is is in any proper subspace defined by $A$, is zero. Use this for the finitely many orthogonal complements of the eigenspaces of $A$.
  $endgroup$
  – Hagen von Eitzen
  Mar 14 at 5:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @HagenvonEitzen: Thanks for your comment. But I could not understand what you mean by "proper subspace defined by $A$"? Do you mean the eigenspaces of $A$? I cannot see how this can be argued. Could you point me a reference?
  $endgroup$
  – MyCindy2012
  Mar 14 at 5:36
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The probability that $A$ can be diagonalized over $mathbbR$ tends to $0$ when $n$ tends to $infty$.
  $endgroup$
  – loup blanc
  Mar 15 at 9:40
  
  

  
  
  
  

add a comment | 

2

$begingroup$

Suppose $v in mathbb R^n$ is a fixed nonzero vector and $A in M_n(mathbb R)$ is a random matrix where each entry is taken from a standard normal distribution over $mathbb R$. We know that the probability $A$ is diagonalizable is $1$. My question is: if we concern those matrices that are diagonalizable (thus with an eigenbasis), what is the probability that those with an eigenbasis and $v$ projects nontrivially on each eigenvector in the eigenbasis, i.e., the probability measure of the set
beginalign*
A in M_n(mathbb R): & texteach entry is generated from a standard normal,\
& textif $A=SLambda S^-1$ (spectral decomposition) then $Sv$ has no zero component.
endalign*

linear-algebra probability-theory random-matrices

share|cite|improve this question

edited Mar 14 at 5:34

MyCindy2012

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Unless $v=0$, the probability that $v$ is is in any proper subspace defined by $A$, is zero. Use this for the finitely many orthogonal complements of the eigenspaces of $A$.
  $endgroup$
  – Hagen von Eitzen
  Mar 14 at 5:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @HagenvonEitzen: Thanks for your comment. But I could not understand what you mean by "proper subspace defined by $A$"? Do you mean the eigenspaces of $A$? I cannot see how this can be argued. Could you point me a reference?
  $endgroup$
  – MyCindy2012
  Mar 14 at 5:36
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The probability that $A$ can be diagonalized over $mathbbR$ tends to $0$ when $n$ tends to $infty$.
  $endgroup$
  – loup blanc
  Mar 15 at 9:40
  
  

  
  
  
  

add a comment | 

$begingroup$

Suppose $v in mathbb R^n$ is a fixed nonzero vector and $A in M_n(mathbb R)$ is a random matrix where each entry is taken from a standard normal distribution over $mathbb R$. We know that the probability $A$ is diagonalizable is $1$. My question is: if we concern those matrices that are diagonalizable (thus with an eigenbasis), what is the probability that those with an eigenbasis and $v$ projects nontrivially on each eigenvector in the eigenbasis, i.e., the probability measure of the set
beginalign*
A in M_n(mathbb R): & texteach entry is generated from a standard normal,\
& textif $A=SLambda S^-1$ (spectral decomposition) then $Sv$ has no zero component.
endalign*

linear-algebra probability-theory random-matrices

share|cite|improve this question

edited Mar 14 at 5:34

MyCindy2012

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211

$endgroup$

share|cite|improve this question

edited Mar 14 at 5:34

MyCindy2012

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211

share|cite|improve this question

edited Mar 14 at 5:34

MyCindy2012

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211

asked Mar 14 at 5:06

MyCindy2012MyCindy2012

9211