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edited Oct 11 2010 at 21:47 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Property (2) implies $p_{ii}=1$ for all $i$, since a positive definite matrix has positive diagonal elements. Property (1) doesn't imply the same thing, since all $p_{ij}$ might be zero (as by Guillaume's comment) [EDIT: a slightly less trivial example is $p_{11}=p_{22}=0$, $p_{12}=p_{21}=1$, and $p_{ij}=\delta_{i,j}$ if $i>2$ or $j>2$ ]. So let us assume $p_{ii}=1$ as part of the hypothesis. Then both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if it is not an equivalence relation, or equivalently (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then,
considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia).
Hence (1),(2),(3) are equivalent, if in (1) one assumes $p_{ii}=1$ for all $i$ (i.e. the symmetric pinching also preserves the diagonal).
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edited Oct 11 2010 at 18:19 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Property (2) implies $p_{ii}=1$ for all $i$, since a positive definite matrix has positive diagonal elements. Property (1) doesn't imply the same thing, since all $p_{ij}$ might be zero (as by Guillaume's comment)comment) [EDIT: a slightly less trivial example is $p_{11}=p_{22}=0$, $p_{12}=p_{21}=1$, and $p_{ij}=\delta_{i,j}$ if $i>2$ or $j>2$ ]. So let us assume $p_{ii}=1$ as part of the hypothesis. Then both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then,
considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia).
Hence (1),(2),(3) are equivalent, if in (1) one assumes $p_{ii}=1$ for all $i$ (i.e. the symmetric pinching also preserves the diagonal).
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edited Oct 11 2010 at 17:10 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Property (2) implies $p_{ii}=1$ for all $i$, since a positive definite matrix has positive diagonal elements. Property (1) doesn't imply the same thing, since all $p_{ij}$ might be zero (as by Guillaume's comment). So let us assume $p_{ii}=1$ as part of the hypothesis. Then both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then,
considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia).
Hence (1),(2),(3) are equivalent, if in (1) one assumes $p_{ii}=1$ for all $i$ (i.e. the symmetric pinching also preserves the diagonal).
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edited Oct 11 2010 at 17:03 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Property (2) implies $p_{ii}=1$ for all $i$, since a positive definite matrix has positive diagonal elements. Property (1) doesn't imply the same thing, since all $p_{ij}$ might be zero (as by Guillaume's comment). So let us assume $p_{ii}=1$ as part of the hypothesis. Then both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then,
considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia).
Hence (1),(2),(3) are equivalent, if one assumes $p_{ii}=1$ for all $i$ (i.e. the symmetric pinching also preserves the diagonal).
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edited Oct 11 2010 at 16:50 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Since
$\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$ has norm $\sqrt{2}$ and $\left(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right)$ has norm $(1+\sqrt{5})/2 > \sqrt{2}$, property Property (1) 2) implies $p_{ii}=1$ for all $i$. Property (2) implies the same thing, i$, since a positive definite matrix has positive diagonal elements. Hence Property (1) doesn't imply the same thing, since all $p_{ij}$ might be zero (as by Guillaume's comment). So let us assume $p_{ii}=1$ as part of the hypothesis. Then both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then,
considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia).
Hence (1),(2),(3) are equivalent.
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edited Oct 11 2010 at 16:40 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Since
$\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$ has norm $\sqrt{2}$ and $\left(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right)$ has norm $(1+\sqrt{5})/2 > \sqrt{2}$, property (1) implies $p_{ii}=1$ for all $i$. Property (2) implies the same thing, since a positive definite matrix has positive diagonal elements. Hence both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then,
considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$, but I'm not sure yetP$. But it is easily seen that $B$ doesn't contract the trace norm (sum of singular values), since the all ones $3\times3$ matrix $E$ has trace norm $3$ and $B(E)$ has trace norm $1+2\sqrt{2}$ (cf Bhatia).
Hence (1),(2),(3) are equivalent.
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edited Oct 11 2010 at 16:18 |
EDIT (partly answering the modified question): Consider a symmetric pinching, $M\mapsto P(M)=(p_{ij}M_{ij})_{ij}$, $p_{ij}=p_{ji}\in\{O,1\}$. Since $\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$ has norm $\sqrt{2}$ and $\left(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right)$ has norm $(1+\sqrt{5})/2 > \sqrt{2}$, property (1) implies $p_{ii}=1$ for all $i$. Property (2) implies the same thing, since a positive definite matrix has positive diagonal elements. Hence both (1) and (2) imply that the relation between indices $\{(i,j): p_{ij}=1\}$ is reflexive and symmetric. If it is not transitive (i.e. if (3) doesn't hold), then there are three distinct indices $i,j,k$ with $p_{ij}=p_{jk}=1$ but $p_{ik}=0$. Let $I=\{i,j,k\}$. Then, considering your $B$ example, $P$ doesn't preserve positive definiteness, since the $I \times I$ principal minor of $P(M)$ can be negative for a positive definite $M$. Hence (2) implies (3), and they are equivalent. Similiarly, restricting $P$ to matrices supported on $I\times I$, if $B$ doesn't contract some "natural" unitarily invariant norm on $3\times 3$ matrices,the same must be true of $P$, but I'm not sure yet.
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answered Oct 11 2010 at 13:11 |
R. Bhatia proved (in Amer. Math. Monthly 107) that the operator $D_k$ taking $M$ to its $k$-th diagonal ($M_{ij}$, $j-i=k$) contracts any unitarily invariant norm, so 1 implies neither of 2, 3. Still, it is an interesting question to characterize those "contractive pinchings".
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