Question on random matrices
$begingroup$
I have $k$ independent and identically distributed matrix valued random variables $X_{1}, ~X_{2}, dots ~X_{k}$. The random matrices are all positive semidefinite with eigenvalues between $0$ and $1$.
I know that $lambda_{1}(mathbb{E}(X_{i})) geq alpha ~forall~ i in [k]$ where $mathbb{E}(~.)$ is the expected value operator and $lambda_{1}(~.~)$ extracts the largest eigenvalue of its argument. $alpha$ is a positive constant (between $0$ and $1$).
I also know that $lambda_{1}(frac{1}{k}sum_{i=1}^{k} X_{i}) geq alpha$ with high probability.
Can we say anything about $lambda_{1} (X_{i})$ for any $i in [k]$? More specifically, can we have a bound on the number of $i$'s for which $lambda_{1}(X_{i}) geq alpha$?
probability-theory probability-distributions random-matrices
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
$endgroup$
add a comment |
$begingroup$
I have $k$ independent and identically distributed matrix valued random variables $X_{1}, ~X_{2}, dots ~X_{k}$. The random matrices are all positive semidefinite with eigenvalues between $0$ and $1$.
I know that $lambda_{1}(mathbb{E}(X_{i})) geq alpha ~forall~ i in [k]$ where $mathbb{E}(~.)$ is the expected value operator and $lambda_{1}(~.~)$ extracts the largest eigenvalue of its argument. $alpha$ is a positive constant (between $0$ and $1$).
I also know that $lambda_{1}(frac{1}{k}sum_{i=1}^{k} X_{i}) geq alpha$ with high probability.
Can we say anything about $lambda_{1} (X_{i})$ for any $i in [k]$? More specifically, can we have a bound on the number of $i$'s for which $lambda_{1}(X_{i}) geq alpha$?
probability-theory probability-distributions random-matrices
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
$endgroup$
add a comment |
$begingroup$
I have $k$ independent and identically distributed matrix valued random variables $X_{1}, ~X_{2}, dots ~X_{k}$. The random matrices are all positive semidefinite with eigenvalues between $0$ and $1$.
I know that $lambda_{1}(mathbb{E}(X_{i})) geq alpha ~forall~ i in [k]$ where $mathbb{E}(~.)$ is the expected value operator and $lambda_{1}(~.~)$ extracts the largest eigenvalue of its argument. $alpha$ is a positive constant (between $0$ and $1$).
I also know that $lambda_{1}(frac{1}{k}sum_{i=1}^{k} X_{i}) geq alpha$ with high probability.
Can we say anything about $lambda_{1} (X_{i})$ for any $i in [k]$? More specifically, can we have a bound on the number of $i$'s for which $lambda_{1}(X_{i}) geq alpha$?
probability-theory probability-distributions random-matrices
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
$endgroup$
I have $k$ independent and identically distributed matrix valued random variables $X_{1}, ~X_{2}, dots ~X_{k}$. The random matrices are all positive semidefinite with eigenvalues between $0$ and $1$.
I know that $lambda_{1}(mathbb{E}(X_{i})) geq alpha ~forall~ i in [k]$ where $mathbb{E}(~.)$ is the expected value operator and $lambda_{1}(~.~)$ extracts the largest eigenvalue of its argument. $alpha$ is a positive constant (between $0$ and $1$).
I also know that $lambda_{1}(frac{1}{k}sum_{i=1}^{k} X_{i}) geq alpha$ with high probability.
Can we say anything about $lambda_{1} (X_{i})$ for any $i in [k]$? More specifically, can we have a bound on the number of $i$'s for which $lambda_{1}(X_{i}) geq alpha$?
probability-theory probability-distributions random-matrices
probability-theory probability-distributions random-matrices
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
asked Jan 17 at 6:38
BlackHat18BlackHat18
11
11
add a comment |
add a comment |
1 Answer
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$begingroup$
Did you work more than $10$' about this question ?
How do you find a random matrix $A$ that is sym. with $spectrum(A)subset [0,1]$ ?
For example, a sample for $Ain M_n$ is as follows:
i) Randomly choose (with uniform distribution on $[0,1]$) $n$ values $(a_i)$.
ii) Randomly choose a skew-sym. matrix $K$ and deduce (via the Cayley formula) an orthogonal matrix $U$.
iii) Put $A=Udiag(a_i)U^T$.
It seems to me that you are interested only by $E(lambda_1(A))$, that is $E(max_i (a_i))$. The answer of this last problem is well-known.
A small effort my friend; come on !
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
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1 Answer
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1 Answer
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active
oldest
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active
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votes
$begingroup$
Did you work more than $10$' about this question ?
How do you find a random matrix $A$ that is sym. with $spectrum(A)subset [0,1]$ ?
For example, a sample for $Ain M_n$ is as follows:
i) Randomly choose (with uniform distribution on $[0,1]$) $n$ values $(a_i)$.
ii) Randomly choose a skew-sym. matrix $K$ and deduce (via the Cayley formula) an orthogonal matrix $U$.
iii) Put $A=Udiag(a_i)U^T$.
It seems to me that you are interested only by $E(lambda_1(A))$, that is $E(max_i (a_i))$. The answer of this last problem is well-known.
A small effort my friend; come on !
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
$endgroup$
add a comment |
$begingroup$
Did you work more than $10$' about this question ?
How do you find a random matrix $A$ that is sym. with $spectrum(A)subset [0,1]$ ?
For example, a sample for $Ain M_n$ is as follows:
i) Randomly choose (with uniform distribution on $[0,1]$) $n$ values $(a_i)$.
ii) Randomly choose a skew-sym. matrix $K$ and deduce (via the Cayley formula) an orthogonal matrix $U$.
iii) Put $A=Udiag(a_i)U^T$.
It seems to me that you are interested only by $E(lambda_1(A))$, that is $E(max_i (a_i))$. The answer of this last problem is well-known.
A small effort my friend; come on !
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
$endgroup$
add a comment |
$begingroup$
Did you work more than $10$' about this question ?
How do you find a random matrix $A$ that is sym. with $spectrum(A)subset [0,1]$ ?
For example, a sample for $Ain M_n$ is as follows:
i) Randomly choose (with uniform distribution on $[0,1]$) $n$ values $(a_i)$.
ii) Randomly choose a skew-sym. matrix $K$ and deduce (via the Cayley formula) an orthogonal matrix $U$.
iii) Put $A=Udiag(a_i)U^T$.
It seems to me that you are interested only by $E(lambda_1(A))$, that is $E(max_i (a_i))$. The answer of this last problem is well-known.
A small effort my friend; come on !
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
$endgroup$
Did you work more than $10$' about this question ?
How do you find a random matrix $A$ that is sym. with $spectrum(A)subset [0,1]$ ?
For example, a sample for $Ain M_n$ is as follows:
i) Randomly choose (with uniform distribution on $[0,1]$) $n$ values $(a_i)$.
ii) Randomly choose a skew-sym. matrix $K$ and deduce (via the Cayley formula) an orthogonal matrix $U$.
iii) Put $A=Udiag(a_i)U^T$.
It seems to me that you are interested only by $E(lambda_1(A))$, that is $E(max_i (a_i))$. The answer of this last problem is well-known.
A small effort my friend; come on !
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
answered Feb 20 at 17:58
loup blancloup blanc
24.3k21852
24.3k21852
add a comment |
add a comment |
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