SVD of a specific matrix and Singular values behaviour
$begingroup$
I have a rectangular matrix $A in mathcal{M}_{l,n} (mathbb{C})$, $l>n$ which has this property :
$$A=left[begin{matrix} M_1 & i M_2 \ M_2 & -i M_1 end{matrix}right]$$
Where $M_i$ are (complex) rectangular matrices with same size $frac l 2 times frac n 2$.
Can i expect (formally) some special behaviour from the singular values of $A$ relatively to its size ?
Some Background :
Numerically, I noticed that the smallest singular values of $A$ are very close to $0$ relatively to the greatest ones when $n < l < 2n$ and not when $l ge 2n$. Then I found out that $A$ has that special structure described at the beginning of that question. I do expect a (quantitative) link between those two things.
My handwavy explanation is that, when $n < l < 2n$, the redundacy of information encoded by the special structure is well captured by the SVD, so we do not need all the eigenvalues to construct a good approximation of $A$.
Thank you.
linear-algebra matrices matrix-decomposition svd singularvalues
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
$endgroup$
add a comment |
$begingroup$
I have a rectangular matrix $A in mathcal{M}_{l,n} (mathbb{C})$, $l>n$ which has this property :
$$A=left[begin{matrix} M_1 & i M_2 \ M_2 & -i M_1 end{matrix}right]$$
Where $M_i$ are (complex) rectangular matrices with same size $frac l 2 times frac n 2$.
Can i expect (formally) some special behaviour from the singular values of $A$ relatively to its size ?
Some Background :
Numerically, I noticed that the smallest singular values of $A$ are very close to $0$ relatively to the greatest ones when $n < l < 2n$ and not when $l ge 2n$. Then I found out that $A$ has that special structure described at the beginning of that question. I do expect a (quantitative) link between those two things.
My handwavy explanation is that, when $n < l < 2n$, the redundacy of information encoded by the special structure is well captured by the SVD, so we do not need all the eigenvalues to construct a good approximation of $A$.
Thank you.
linear-algebra matrices matrix-decomposition svd singularvalues
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
$endgroup$
add a comment |
$begingroup$
I have a rectangular matrix $A in mathcal{M}_{l,n} (mathbb{C})$, $l>n$ which has this property :
$$A=left[begin{matrix} M_1 & i M_2 \ M_2 & -i M_1 end{matrix}right]$$
Where $M_i$ are (complex) rectangular matrices with same size $frac l 2 times frac n 2$.
Can i expect (formally) some special behaviour from the singular values of $A$ relatively to its size ?
Some Background :
Numerically, I noticed that the smallest singular values of $A$ are very close to $0$ relatively to the greatest ones when $n < l < 2n$ and not when $l ge 2n$. Then I found out that $A$ has that special structure described at the beginning of that question. I do expect a (quantitative) link between those two things.
My handwavy explanation is that, when $n < l < 2n$, the redundacy of information encoded by the special structure is well captured by the SVD, so we do not need all the eigenvalues to construct a good approximation of $A$.
Thank you.
linear-algebra matrices matrix-decomposition svd singularvalues
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
$endgroup$
I have a rectangular matrix $A in mathcal{M}_{l,n} (mathbb{C})$, $l>n$ which has this property :
$$A=left[begin{matrix} M_1 & i M_2 \ M_2 & -i M_1 end{matrix}right]$$
Where $M_i$ are (complex) rectangular matrices with same size $frac l 2 times frac n 2$.
Can i expect (formally) some special behaviour from the singular values of $A$ relatively to its size ?
Some Background :
Numerically, I noticed that the smallest singular values of $A$ are very close to $0$ relatively to the greatest ones when $n < l < 2n$ and not when $l ge 2n$. Then I found out that $A$ has that special structure described at the beginning of that question. I do expect a (quantitative) link between those two things.
My handwavy explanation is that, when $n < l < 2n$, the redundacy of information encoded by the special structure is well captured by the SVD, so we do not need all the eigenvalues to construct a good approximation of $A$.
Thank you.
linear-algebra matrices matrix-decomposition svd singularvalues
linear-algebra matrices matrix-decomposition svd singularvalues
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
asked Dec 14 '18 at 15:04
nicomezinicomezi
4,2141920
4,2141920
add a comment |
add a comment |
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