Growth of “spectral bounded” sequences of products of matrices

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I am trying to proof the following, of which I am sure it is true. I would be pleased if you could give me hints, how to approach such a problem.

Given a finite set of matrices $mathcalA=A_jinmathbbR^stimes s,j=1,ldots,J$. Define the set
$$
mathcalM=
(A_i_ncdots A_i_1)_ninmathcalA^mathbbN
:
%i_jin1,ldots,J,
rho(A_i_ncdots A_i_1)<1
text and
A_i_n-1cdots A_i_1inmathcalM
.
$$

Thus, $mathcalM$ consists of sequences of products, all of whose entries have spectral radius then less then one.

Show that there exists $C>0$ such that for all $xinmathbbR^s$ and all sequences $(A_i_ncdots A_i_1)_ninmathbbNinmathcalM$

$$
sup_ninmathbbN |A_i_ncdots A_i_1x|<C.
$$

---

EDIT: After more thoughts about the problem, and making some numerical experiments, the above would follow from the following generalization of Gelfands formula, which I suppose is true.

Let $(i_n)_nin1,ldots,J$. Then for $(B_n)_n=(A_i_ncdots A_i_1)_n$,
$$
lim_nrightarrowinfty left(fracrho(B_n)B_nright)^1/n=1
$$

Clearly $rho(M_n)leq|M_n|$, but the other direction I could not proof yet.

sequences-and-series matrices spectral-radius

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edited Jul 26 at 11:43

asked Jul 25 at 13:32

tommsch

1387

add a comment | 

up vote
1
down vote

favorite

I am trying to proof the following, of which I am sure it is true. I would be pleased if you could give me hints, how to approach such a problem.

Given a finite set of matrices $mathcalA=A_jinmathbbR^stimes s,j=1,ldots,J$. Define the set
$$
mathcalM=
(A_i_ncdots A_i_1)_ninmathcalA^mathbbN
:
%i_jin1,ldots,J,
rho(A_i_ncdots A_i_1)<1
text and
A_i_n-1cdots A_i_1inmathcalM
.
$$

Thus, $mathcalM$ consists of sequences of products, all of whose entries have spectral radius then less then one.

Show that there exists $C>0$ such that for all $xinmathbbR^s$ and all sequences $(A_i_ncdots A_i_1)_ninmathbbNinmathcalM$

$$
sup_ninmathbbN |A_i_ncdots A_i_1x|<C.
$$

---

EDIT: After more thoughts about the problem, and making some numerical experiments, the above would follow from the following generalization of Gelfands formula, which I suppose is true.

Let $(i_n)_nin1,ldots,J$. Then for $(B_n)_n=(A_i_ncdots A_i_1)_n$,
$$
lim_nrightarrowinfty left(fracrho(B_n)B_nright)^1/n=1
$$

Clearly $rho(M_n)leq|M_n|$, but the other direction I could not proof yet.

sequences-and-series matrices spectral-radius

share|cite|improve this question

edited Jul 26 at 11:43

asked Jul 25 at 13:32

tommsch

1387

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

I am trying to proof the following, of which I am sure it is true. I would be pleased if you could give me hints, how to approach such a problem.

Given a finite set of matrices $mathcalA=A_jinmathbbR^stimes s,j=1,ldots,J$. Define the set
$$
mathcalM=
(A_i_ncdots A_i_1)_ninmathcalA^mathbbN
:
%i_jin1,ldots,J,
rho(A_i_ncdots A_i_1)<1
text and
A_i_n-1cdots A_i_1inmathcalM
.
$$

Thus, $mathcalM$ consists of sequences of products, all of whose entries have spectral radius then less then one.

Show that there exists $C>0$ such that for all $xinmathbbR^s$ and all sequences $(A_i_ncdots A_i_1)_ninmathbbNinmathcalM$

$$
sup_ninmathbbN |A_i_ncdots A_i_1x|<C.
$$

---

EDIT: After more thoughts about the problem, and making some numerical experiments, the above would follow from the following generalization of Gelfands formula, which I suppose is true.

Let $(i_n)_nin1,ldots,J$. Then for $(B_n)_n=(A_i_ncdots A_i_1)_n$,
$$
lim_nrightarrowinfty left(fracrho(B_n)B_nright)^1/n=1
$$

Clearly $rho(M_n)leq|M_n|$, but the other direction I could not proof yet.

sequences-and-series matrices spectral-radius

share|cite|improve this question

edited Jul 26 at 11:43

asked Jul 25 at 13:32

tommsch

1387

I am trying to proof the following, of which I am sure it is true. I would be pleased if you could give me hints, how to approach such a problem.

Given a finite set of matrices $mathcalA=A_jinmathbbR^stimes s,j=1,ldots,J$. Define the set
$$
mathcalM=
(A_i_ncdots A_i_1)_ninmathcalA^mathbbN
:
%i_jin1,ldots,J,
rho(A_i_ncdots A_i_1)<1
text and
A_i_n-1cdots A_i_1inmathcalM
.
$$

Thus, $mathcalM$ consists of sequences of products, all of whose entries have spectral radius then less then one.

Show that there exists $C>0$ such that for all $xinmathbbR^s$ and all sequences $(A_i_ncdots A_i_1)_ninmathbbNinmathcalM$

$$
sup_ninmathbbN |A_i_ncdots A_i_1x|<C.
$$

---

EDIT: After more thoughts about the problem, and making some numerical experiments, the above would follow from the following generalization of Gelfands formula, which I suppose is true.

Let $(i_n)_nin1,ldots,J$. Then for $(B_n)_n=(A_i_ncdots A_i_1)_n$,
$$
lim_nrightarrowinfty left(fracrho(B_n)B_nright)^1/n=1
$$

Clearly $rho(M_n)leq|M_n|$, but the other direction I could not proof yet.

sequences-and-series matrices spectral-radius

share|cite|improve this question

edited Jul 26 at 11:43

asked Jul 25 at 13:32

tommsch

1387

share|cite|improve this question

share|cite|improve this question

edited Jul 26 at 11:43

edited Jul 26 at 11:43

edited Jul 26 at 11:43

asked Jul 25 at 13:32

tommsch

1387

asked Jul 25 at 13:32

tommsch

1387

asked Jul 25 at 13:32

tommsch

1387

1387

add a comment | 

add a comment | 

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