Linear maps on matrices preserving parallel pairs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914911182913536 |
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| author | Li, Chi-Kwong Tsai, Ming-Cheng Wang, Ya-Shu Wong, Ngai-Ching |
| author_facet | Li, Chi-Kwong Tsai, Ming-Cheng Wang, Ya-Shu Wong, Ngai-Ching |
| contents | Two (real or complex) $m\times n$ matrices $A$ and $B$ are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm $\|\cdot\|$ if $\|A+ μB\| = \|A\| + \|B\|$ for some scalar $μ$ with $|μ|=1$ (resp. $μ=1$). We study linear maps $T$ on $m\times n$ matrices preserving parallel (resp. TEA) pairs, i.e., $T(A)$ and $T(B)$ are parallel (resp. TEA) whenever $A$ and $B$ are parallel (resp. TEA).
It is shown that when $m,n \ge 2$ and $(m,n) \ne (2,2)$, a nonzero linear map $T$ preserving TEA pairs if and only if it is a positive multiple of a linear isometry, namely, $T$ has the form
$$(1) \quad A \mapsto γUAV \quad \quad \text{or} \quad \quad (2) \quad A \mapsto γUA^{t} V \quad (\text{in this case}, m = n),$$ for a positive number $γ$, and unitary (or real orthogonal) matrices $U$ and $V$ of appropriate sizes. Linear maps preserving parallel pairs are those carrying form (1), (2), or the form
$$ (3) \ A \mapsto f(A) Z$$
for a linear functional $f$ and a fixed matrix $Z$.
The case when $(m,n) = (2,2)$ is more complicated. There are linear maps of $2\times 2$ matrices preserving parallel pairs or TEA pairs neither of the form (1), (2) nor (3) above. Complete characterization of such maps is given with some intricate computation and techniques in matrix groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear maps on matrices preserving parallel pairs Li, Chi-Kwong Tsai, Ming-Cheng Wang, Ya-Shu Wong, Ngai-Ching Rings and Algebras 15A86, 15A60 Two (real or complex) $m\times n$ matrices $A$ and $B$ are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm $\|\cdot\|$ if $\|A+ μB\| = \|A\| + \|B\|$ for some scalar $μ$ with $|μ|=1$ (resp. $μ=1$). We study linear maps $T$ on $m\times n$ matrices preserving parallel (resp. TEA) pairs, i.e., $T(A)$ and $T(B)$ are parallel (resp. TEA) whenever $A$ and $B$ are parallel (resp. TEA). It is shown that when $m,n \ge 2$ and $(m,n) \ne (2,2)$, a nonzero linear map $T$ preserving TEA pairs if and only if it is a positive multiple of a linear isometry, namely, $T$ has the form $$(1) \quad A \mapsto γUAV \quad \quad \text{or} \quad \quad (2) \quad A \mapsto γUA^{t} V \quad (\text{in this case}, m = n),$$ for a positive number $γ$, and unitary (or real orthogonal) matrices $U$ and $V$ of appropriate sizes. Linear maps preserving parallel pairs are those carrying form (1), (2), or the form $$ (3) \ A \mapsto f(A) Z$$ for a linear functional $f$ and a fixed matrix $Z$. The case when $(m,n) = (2,2)$ is more complicated. There are linear maps of $2\times 2$ matrices preserving parallel pairs or TEA pairs neither of the form (1), (2) nor (3) above. Complete characterization of such maps is given with some intricate computation and techniques in matrix groups. |
| title | Linear maps on matrices preserving parallel pairs |
| topic | Rings and Algebras 15A86, 15A60 |
| url | https://arxiv.org/abs/2408.06366 |