Eigenvalues of a multinomial covariance matrix
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The following matrix shows up in studying multinomial distributions as the covariance matrix. Let $p$ be a column vector of dimension $k$ with $p_igeq 0, sum_{i=1}^k p_i = 1.$ Let
$$A:=mathrm{Diag}(p) - pp^T,$$
where $mathrm{Diag}(p)$ is a diagonal matrix with $p$ on the diagonal.
$A$ is a positive semidefinite matrix. One of its eigenvalues is zero (corresponding to the all-ones eigenvector). What are its other eigenvalues as a function of $p$? Is there a closed-form expression for those eigenvalues?
linear-algebra
asked Oct 8 '15 at 19:23
Hedonist
640312
add a comment |
up vote
3
down vote
favorite
The following matrix shows up in studying multinomial distributions as the covariance matrix. Let $p$ be a column vector of dimension $k$ with $p_igeq 0, sum_{i=1}^k p_i = 1.$ Let
$$A:=mathrm{Diag}(p) - pp^T,$$
where $mathrm{Diag}(p)$ is a diagonal matrix with $p$ on the diagonal.
$A$ is a positive semidefinite matrix. One of its eigenvalues is zero (corresponding to the all-ones eigenvector). What are its other eigenvalues as a function of $p$? Is there a closed-form expression for those eigenvalues?
linear-algebra
asked Oct 8 '15 at 19:23
Hedonist
640312
Maybe relevant: math.stackexchange.com/questions/730134/… math.stackexchange.com/questions/506761/…
– Jean-Claude Arbaut
Oct 8 '15 at 19:36
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
The following matrix shows up in studying multinomial distributions as the covariance matrix. Let $p$ be a column vector of dimension $k$ with $p_igeq 0, sum_{i=1}^k p_i = 1.$ Let
$$A:=mathrm{Diag}(p) - pp^T,$$
where $mathrm{Diag}(p)$ is a diagonal matrix with $p$ on the diagonal.
$A$ is a positive semidefinite matrix. One of its eigenvalues is zero (corresponding to the all-ones eigenvector). What are its other eigenvalues as a function of $p$? Is there a closed-form expression for those eigenvalues?
linear-algebra
asked Oct 8 '15 at 19:23
Hedonist
640312
The following matrix shows up in studying multinomial distributions as the covariance matrix. Let $p$ be a column vector of dimension $k$ with $p_igeq 0, sum_{i=1}^k p_i = 1.$ Let
$$A:=mathrm{Diag}(p) - pp^T,$$
where $mathrm{Diag}(p)$ is a diagonal matrix with $p$ on the diagonal.
$A$ is a positive semidefinite matrix. One of its eigenvalues is zero (corresponding to the all-ones eigenvector). What are its other eigenvalues as a function of $p$? Is there a closed-form expression for those eigenvalues?
linear-algebra
linear-algebra
asked Oct 8 '15 at 19:23
Hedonist
640312
asked Oct 8 '15 at 19:23
Hedonist
640312
asked Oct 8 '15 at 19:23
Hedonist
640312
asked Oct 8 '15 at 19:23
Hedonist
640312
asked Oct 8 '15 at 19:23
Hedonist
640312
640312
Maybe relevant: math.stackexchange.com/questions/730134/… math.stackexchange.com/questions/506761/…
– Jean-Claude Arbaut
Oct 8 '15 at 19:36
add a comment |
Maybe relevant: math.stackexchange.com/questions/730134/… math.stackexchange.com/questions/506761/…
– Jean-Claude Arbaut
Oct 8 '15 at 19:36
Maybe relevant: math.stackexchange.com/questions/730134/… math.stackexchange.com/questions/506761/…
– Jean-Claude Arbaut
Oct 8 '15 at 19:36
Maybe relevant: math.stackexchange.com/questions/730134/… math.stackexchange.com/questions/506761/…
– Jean-Claude Arbaut
Oct 8 '15 at 19:36
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
In general no closed form. See this paper:
https://projecteuclid.org/download/pdfview_1/euclid.bjps/1405603508
edited Nov 16 at 14:04
amWhy
191k27223437
answered Nov 16 at 2:21
Mingyue Gao
362
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
In general no closed form. See this paper:
https://projecteuclid.org/download/pdfview_1/euclid.bjps/1405603508
edited Nov 16 at 14:04
amWhy
191k27223437
answered Nov 16 at 2:21
Mingyue Gao
362
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
add a comment |
up vote
2
down vote
accepted
In general no closed form. See this paper:
https://projecteuclid.org/download/pdfview_1/euclid.bjps/1405603508
edited Nov 16 at 14:04
amWhy
191k27223437
answered Nov 16 at 2:21
Mingyue Gao
362
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
In general no closed form. See this paper:
https://projecteuclid.org/download/pdfview_1/euclid.bjps/1405603508
edited Nov 16 at 14:04
amWhy
191k27223437
answered Nov 16 at 2:21
Mingyue Gao
362
In general no closed form. See this paper:
https://projecteuclid.org/download/pdfview_1/euclid.bjps/1405603508
edited Nov 16 at 14:04
amWhy
191k27223437
answered Nov 16 at 2:21
Mingyue Gao
362
edited Nov 16 at 14:04
amWhy
191k27223437
edited Nov 16 at 14:04
amWhy
191k27223437
edited Nov 16 at 14:04
amWhy
191k27223437
191k27223437
answered Nov 16 at 2:21
Mingyue Gao
362
answered Nov 16 at 2:21
Mingyue Gao
362
answered Nov 16 at 2:21
Mingyue Gao
362
362
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
add a comment |
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
Link-only answer (without at least some elaboration) is generally unacceptable. Either you should add some substantial descriptions, or you can wait till you have enough reputation to leave a link-only comment (which is perfectly fine).
– Lee David Chung Lin
Nov 16 at 2:54
add a comment |
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Maybe relevant: math.stackexchange.com/questions/730134/… math.stackexchange.com/questions/506761/…
– Jean-Claude Arbaut
Oct 8 '15 at 19:36