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Trace of squared non-square matrix
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In reading a paper I came across this expression which I don't quite understand:
$$
fraclambda_12Noperatornametrleft((mathbfH^MmathbfH^M)^Tright)
$$
For context, $lambda_1$ and $N$ are scalars and $mathbfH^M$ ($H$ henceforth) is a $MxN$ matrix.
The paper claims that this value is related to the variance of the column vectors which make up H, but either there is a typo or (quite likely) a linear algebra concept I don't know. The reason I am confused is that trace requires square input, and if H were to be made square then the transpose would have to apply to one of the two $H$s, not both after multiplying them (which itself doesn't make sense to me because only a square matrix can be multiplied by itself in the first place).
The paper is here, the mentioned expression is at the bottom of the third page in equation (5).
linear-algebra matrices trace
asked Jul 25 at 18:28
Sean Hastings
32
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up vote
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down vote
favorite
In reading a paper I came across this expression which I don't quite understand:
$$
fraclambda_12Noperatornametrleft((mathbfH^MmathbfH^M)^Tright)
$$
For context, $lambda_1$ and $N$ are scalars and $mathbfH^M$ ($H$ henceforth) is a $MxN$ matrix.
The paper claims that this value is related to the variance of the column vectors which make up H, but either there is a typo or (quite likely) a linear algebra concept I don't know. The reason I am confused is that trace requires square input, and if H were to be made square then the transpose would have to apply to one of the two $H$s, not both after multiplying them (which itself doesn't make sense to me because only a square matrix can be multiplied by itself in the first place).
The paper is here, the mentioned expression is at the bottom of the third page in equation (5).
linear-algebra matrices trace
asked Jul 25 at 18:28
Sean Hastings
32
Could it be $mathbfH^M (mathbfH^M)^T$ instead? That would be square for any matrix $mathbfH^M$.
– Matthew Leingang
Jul 25 at 18:36
That is what I had been thinking, but assuming that I haven't been able to figure out what the relationship to the variance of the column vectors would be. At that point it becomes a stats question more than a linalg one. If that were a typo then the trace results in the sum of the element-wise square of H, which to me seems more correlated to magnitude of values than variance.
– Sean Hastings
Jul 25 at 18:40
Good point, added.
– Sean Hastings
Jul 25 at 18:44
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In reading a paper I came across this expression which I don't quite understand:
$$
fraclambda_12Noperatornametrleft((mathbfH^MmathbfH^M)^Tright)
$$
For context, $lambda_1$ and $N$ are scalars and $mathbfH^M$ ($H$ henceforth) is a $MxN$ matrix.
The paper claims that this value is related to the variance of the column vectors which make up H, but either there is a typo or (quite likely) a linear algebra concept I don't know. The reason I am confused is that trace requires square input, and if H were to be made square then the transpose would have to apply to one of the two $H$s, not both after multiplying them (which itself doesn't make sense to me because only a square matrix can be multiplied by itself in the first place).
The paper is here, the mentioned expression is at the bottom of the third page in equation (5).
linear-algebra matrices trace
asked Jul 25 at 18:28
Sean Hastings
32
In reading a paper I came across this expression which I don't quite understand:
$$
fraclambda_12Noperatornametrleft((mathbfH^MmathbfH^M)^Tright)
$$
For context, $lambda_1$ and $N$ are scalars and $mathbfH^M$ ($H$ henceforth) is a $MxN$ matrix.
The paper claims that this value is related to the variance of the column vectors which make up H, but either there is a typo or (quite likely) a linear algebra concept I don't know. The reason I am confused is that trace requires square input, and if H were to be made square then the transpose would have to apply to one of the two $H$s, not both after multiplying them (which itself doesn't make sense to me because only a square matrix can be multiplied by itself in the first place).
The paper is here, the mentioned expression is at the bottom of the third page in equation (5).
linear-algebra matrices trace
asked Jul 25 at 18:28
Sean Hastings
32
edited Jul 25 at 18:44
asked Jul 25 at 18:28
Sean Hastings
32
asked Jul 25 at 18:28
Sean Hastings
32
asked Jul 25 at 18:28
Sean Hastings
32
32
Could it be $mathbfH^M (mathbfH^M)^T$ instead? That would be square for any matrix $mathbfH^M$.
– Matthew Leingang
Jul 25 at 18:36
That is what I had been thinking, but assuming that I haven't been able to figure out what the relationship to the variance of the column vectors would be. At that point it becomes a stats question more than a linalg one. If that were a typo then the trace results in the sum of the element-wise square of H, which to me seems more correlated to magnitude of values than variance.
– Sean Hastings
Jul 25 at 18:40
Good point, added.
– Sean Hastings
Jul 25 at 18:44
add a comment |Â
Could it be $mathbfH^M (mathbfH^M)^T$ instead? That would be square for any matrix $mathbfH^M$.
– Matthew Leingang
Jul 25 at 18:36
That is what I had been thinking, but assuming that I haven't been able to figure out what the relationship to the variance of the column vectors would be. At that point it becomes a stats question more than a linalg one. If that were a typo then the trace results in the sum of the element-wise square of H, which to me seems more correlated to magnitude of values than variance.
– Sean Hastings
Jul 25 at 18:40
Good point, added.
– Sean Hastings
Jul 25 at 18:44
Could it be $mathbfH^M (mathbfH^M)^T$ instead? That would be square for any matrix $mathbfH^M$.
– Matthew Leingang
Jul 25 at 18:36
Could it be $mathbfH^M (mathbfH^M)^T$ instead? That would be square for any matrix $mathbfH^M$.
– Matthew Leingang
Jul 25 at 18:36
That is what I had been thinking, but assuming that I haven't been able to figure out what the relationship to the variance of the column vectors would be. At that point it becomes a stats question more than a linalg one. If that were a typo then the trace results in the sum of the element-wise square of H, which to me seems more correlated to magnitude of values than variance.
– Sean Hastings
Jul 25 at 18:40
That is what I had been thinking, but assuming that I haven't been able to figure out what the relationship to the variance of the column vectors would be. At that point it becomes a stats question more than a linalg one. If that were a typo then the trace results in the sum of the element-wise square of H, which to me seems more correlated to magnitude of values than variance.
– Sean Hastings
Jul 25 at 18:40
Good point, added.
– Sean Hastings
Jul 25 at 18:44
Good point, added.
– Sean Hastings
Jul 25 at 18:44
add a comment |Â
1 Answer
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It looks like a typo. In the paper that you have linked, the equation(12) on page(4) rewrites
$$
arg.min .J = ... -fraclambda_12Bigl(trbigl(frac1NH^M(H^M)^Tbigr)+alpha tr(Sigma_B-Sigma_W)Bigr) + ...
$$
with the transpose correctly placed.
answered Jul 26 at 15:52
artha
935
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
It looks like a typo. In the paper that you have linked, the equation(12) on page(4) rewrites
$$
arg.min .J = ... -fraclambda_12Bigl(trbigl(frac1NH^M(H^M)^Tbigr)+alpha tr(Sigma_B-Sigma_W)Bigr) + ...
$$
with the transpose correctly placed.
answered Jul 26 at 15:52
artha
935
add a comment |Â
up vote
0
down vote
accepted
It looks like a typo. In the paper that you have linked, the equation(12) on page(4) rewrites
$$
arg.min .J = ... -fraclambda_12Bigl(trbigl(frac1NH^M(H^M)^Tbigr)+alpha tr(Sigma_B-Sigma_W)Bigr) + ...
$$
with the transpose correctly placed.
answered Jul 26 at 15:52
artha
935
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
It looks like a typo. In the paper that you have linked, the equation(12) on page(4) rewrites
$$
arg.min .J = ... -fraclambda_12Bigl(trbigl(frac1NH^M(H^M)^Tbigr)+alpha tr(Sigma_B-Sigma_W)Bigr) + ...
$$
with the transpose correctly placed.
answered Jul 26 at 15:52
artha
935
It looks like a typo. In the paper that you have linked, the equation(12) on page(4) rewrites
$$
arg.min .J = ... -fraclambda_12Bigl(trbigl(frac1NH^M(H^M)^Tbigr)+alpha tr(Sigma_B-Sigma_W)Bigr) + ...
$$
with the transpose correctly placed.
answered Jul 26 at 15:52
artha
935
answered Jul 26 at 15:52
artha
935
answered Jul 26 at 15:52
artha
935
answered Jul 26 at 15:52
artha
935
935
add a comment |Â
add a comment |Â
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Could it be $mathbfH^M (mathbfH^M)^T$ instead? That would be square for any matrix $mathbfH^M$.
– Matthew Leingang
Jul 25 at 18:36
That is what I had been thinking, but assuming that I haven't been able to figure out what the relationship to the variance of the column vectors would be. At that point it becomes a stats question more than a linalg one. If that were a typo then the trace results in the sum of the element-wise square of H, which to me seems more correlated to magnitude of values than variance.
– Sean Hastings
Jul 25 at 18:40
Good point, added.
– Sean Hastings
Jul 25 at 18:44