Yoshino, Takashi

Kyoto University Research Information Repository

TitleNormal Toeplitz matrices on $\mathbb{C}^n$ (OperatorInequalities and Related Area)Author(s)Yoshino, TakashiCitation数理解析研究所講究録 (2000), 1144: 64-69Issue Date2000-04URL http://hdl.handle.net/2433/63916RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityNormal Toeplitz matrices on $\mathbb{C}^{n}$(Takashi Yoshino)In this lecture, we gave an another proof of the characterization given in[1] and also in [2] of the normal Toeplitz matrices on $\mathbb{C}^{n}$ .For any positive integer $n$ and for any $a_{j,k}\in \mathbb{C}(j, k=1_{0}.2, \cdots, n)$ , let$A==(a_{j,k})$ on $\mathbb{C}^{n}$and letThen$WAW=,$ (1)Let $tA=$ ($a_{n+1-k,n}$ $1-j$ ) and let ${}^{t}A=(a_{k,j})$ . Then, by (1), we have$WAW={}^{t}(_{t}A)=_{t}(^{t}A)$ (2)and $t(AB)(2)={}^{t}(\mathrm{T},\mathrm{f}^{f}ABW)=\{t(WAW)(WBW)\}$$={}^{t}(1/\mathrm{f}’B\mathfrak{s},V){}^{t}(WAW)=tBtA$. (3)(2)1144 2000 64-69 64If $T$ is a Toeplitz matrix on $\mathbb{C}^{n}$ , then$tT=T$ (4)and $t(T^{*}T)=tTt(T^{*})=TT^{*}$ (5)(3) (4)and, in particular, we have$T^{*}T$ ; Toeplitz matrix $arrow$ $t(\tau*\tau)=T^{*}T$(4)$\frac{arrow}{(5)}$T,. normalSince $t(T^{*}T-\tau T*)=t(\tau*T)-t(\tau\tau*)=TT^{*}-T^{*}\tau$ , we have the $\dot{\mathrm{f}}\mathrm{o}\mathrm{l}1_{\mathrm{o}\mathrm{W}}\mathrm{i}\mathrm{n}\mathrm{g}$(5)by (4).Theorem 1. For a Toeplitz matrix $T$ on $\mathbb{C}^{n},$ $T$ is normal if and only if$T^{*}T-^{\tau\tau*}$ is a Toeplitz matrix.Henceforth we assume that the Toeplitz matrix $T$ on $\mathbb{C}^{n}$ is represented asfollows ;$T=$where $a_{j}$ , $(j=0, \pm 1, \pm 2, \cdots, \pm(n-1))$ are some complex numbers. Then$T^{*}T=(t_{j,k})$ , where $t_{j,k}= \sum_{1m=}^{?\iota}\overline{a}am-jm-k$ , and $T^{*}T-\tau T^{*}=(\alpha_{j,k})$ , where$\alpha_{j,k}=\sum_{m=1}^{n}\overline{a_{m}-j}a_{m}-k-\sum_{m=1}^{7\iota}\overline{ak-?n}aj-m$and hence, by Theorem 1, we have the following.Theorem 2. The Toeplitz matrix $T$ on $\mathbb{C}^{?b}$ is normal if and only if, forany $j,$ $k=1,2,$ $\cdots,$ $n-1$ ,$\overline{a_{-j}}a_{-k}-\overline{ak}a_{j}=\overline{a_{n-j}}a_{n-k}-\overline{a_{-(n-k)^{a_{-}}(}}n-j)$ . (6)65Let $n-k$ instead of $k$ in (6). Then, for any $j,$ $k=1,2,$ $\cdots,$ $n-1$ , we have$\overline{a_{-j}}a_{-(k)}n--\overline{a_{n-k}}a_{j}=\overline{a_{n-j}}a_{k}-\overline{a_{-k}}a_{-(n-}j)$. (6-1)Let $k=j$ in (6) and (6-1) respectively Then, for any $j=12\text{ }\cdots,$ $n-1$ ,we have$|a_{-j}|^{2}-|a_{j}|^{2}=|a_{r\iota-j}|^{\mathit{2}}-|a_{-(_{?t-}j})|^{2}$ (6-2)and $\overline{a_{-j}}a_{-(_{\mathit{7}\mathrm{L}-}j)}=\overline{a_{?\iota-j}}a_{j}$ (6-3)and hence $|a_{-j}|=|a_{n-j}|$ or $|a_{-j}|=|a_{j}|$i.e., $\{$$a_{-j}=e^{\iota\varphi_{j}}\overline{a_{j}}$or $a_{-j}=e^{i\psi_{j}}an-j$for some $\varphi_{j},$ $\psi_{j}\in[0,2\pi)$ . (6-4)If $a_{-j}=e^{\grave{l}}\varphi_{j}\overline{a_{j}}$, then, by (6-3), we have$0=e^{-i\varphi}ajja-(n-j)-\overline{c\iota_{n-jj}}a=e^{-i\varphi_{j}}a_{j}(a_{-(_{\mathit{7}\iota-})}j-e^{i\varphi_{j}}\overline{a_{n-}j})$and$a_{-j}=e^{i\varphi_{j}}\overline{a_{j}}(c\iota_{j}\neq 0)$ implies $a_{-(\tau\iota-j)}=e^{i\varphi_{j}}\overline{a_{?\iota-j}}$ (6-5)And if $a_{-j}=e^{i\psi_{j}}an-j$ ’ then, by (6-3), we have$0=e^{-i\psi_{j}}\overline{a?\iota-j}a-(n-j)-\overline{a_{n-j}}a_{j}=e^{-i\psi_{j}}\overline{a_{n-}j}(a-(_{7\iota-}j)-e^{i\psi_{j}}(\iota_{j})$and$a_{-j}=e^{i\psi_{j}}an-j(a_{n-j}\neq 0)$ implies $a_{-(n-j)}=e^{i\psi_{j}}a_{j}$ (6-6)In particular, if $\{$$a_{-j}=e^{x\varphi j}\overline{a_{j}}$ $(a_{j}\neq 0)$then, by (6), (6-5) and$a_{-k}=e^{i\psi_{k}}a_{n-}k$ $(a_{n-k}\neq 0)$’(6-6), we have$\{$$a_{-j}=e^{i\varphi_{j}}\overline{a_{j}}=e^{i\varphi_{j}}e^{-i(\varphi j}-^{\psi_{k}})_{a_{n-}}j=e^{i\psi_{k}}a_{n-j}$Or $a_{-k}=e^{i\psi_{k}}a_{n-}k=e^{i\psi_{k}}e-i(\psi k^{-\varphi}’)\overline{a_{k}}=e^{i\varphi_{\mathrm{J}}}.\overline{a_{k}}$66and$\{$$a_{-j}=e^{i\varphi_{j}}\overline{a_{j}}$ $(a_{j}\neq 0)$$a_{-k}=e^{i\psi_{k}}a_{n^{-}}k$ $(a_{n-k}\neq 0)$implies $\{$$a_{-j}=e^{i\psi_{k}}a_{n^{-}}j$or $a_{-k}=e^{i\varphi_{j}}\overline{a_{k}}$.(6-7)Lemma 1. Under the condition (6), if $a_{j}\neq 0,$ $a_{k}\neq 0,$ $a_{n-j}\neq 0,$ $a_{n-k}\neq 0$and if $| \frac{a_{r\iota-j}}{a_{j}}|\neq|\frac{a_{\tau\iota-k}}{a_{k}}|$ , then, for some $\varphi,$ $\psi\in[0,2\pi)$ ,$\{$$a_{-j}=e^{i\varphi}\overline{a_{j}}$$a_{-k}=e^{i\varphi}\overline{a_{k}}$or $\{$$a_{-j}--e^{i\psi_{a_{n-j}}}$$a_{-k}=e^{i\psi}a_{n-k}$ .Lemma 2. Under the condition (6), if $a_{j}\neq 0,$ $a_{k}\neq 0,$ $a_{n-j}\neq 0,$ $a_{n-k}..\neq 0$and if $\overline{\frac{a_{n-j}}{a_{j}}}\neq\overline{\frac{a_{n-k}}{a_{k}}}$ , then$\{$$a_{-j}=e^{i\varphi}\overline{a_{j}}$$c\iota_{-k}=‘\supset i\varphi_{\overline{\zeta}}\iota_{k}$$(\mathrm{a}\mathrm{n}\mathrm{d}$ akoimplies that, for any $m=\perp,$ $2,$ $\cdots,$ $n-1$ ,$a_{-m}=e^{i\varphi}\overline{a_{m}}$ (and $a_{-m}=e^{i\psi}a_{n-m}$ respectively).Lemma 3. Under the condition (6), if $a_{j}\neq 0,$ $a_{k}\neq 0,$ $a_{n-j}\neq 0_{c}.a_{n-k}\neq 0$and if $| \frac{a_{r\iota-j}}{a_{j}}|=|\frac{a_{n-k}}{a_{k}}|=|\frac{a_{k}}{a_{r\iota-k}}|$ , then$\{$$a_{-j}=e^{i\varphi_{j}}\overline{aj}=e^{i\psi}ja_{n}-j$$a_{-k}=e\overline{a_{k}}=i\varphi_{k}e^{i}\psi ka_{n-}k$ ,$\{$$a_{n-j}=e^{i(\varphi_{j}}-\psi_{j})\overline{a_{j}}$$a_{n-k}=e^{i(\psi}\varphi_{k}-k)\overline{a_{k}}$and $\{$$\varphi_{j}=\varphi_{k}$or $\psi_{j}=\psi_{k}$ .In particular, if $\frac{a_{n-j}}{\overline{a_{j}}}=\frac{a_{n-k}}{\overline{a_{k}}}$ , then$\{$$a_{-j}=e^{i\varphi_{j}}\overline{a_{j}}=e^{i\tau \mathit{1})}ja_{n-}j$$a_{-k}=e^{i\varphi_{j}}\overline{a_{k}}=e^{i\psi}ja_{\mathit{7}\iota-k}$ .Lemma 4. Under the condition (6), if $a_{-j}=e^{i\varphi}\overline{a_{j}}(a_{j}\neq 0)$ and if $a_{n-j}=0$ ,then$a_{-m}=e^{i\varphi}\overline{a_{m}}(m=1,2, \cdots, n-1)$ .67Lemma 5. Under the condition (6), if $a_{-j}=e^{i\psi}a_{n-j}(a_{n-j}\neq 0)$ and if$a_{j}=0$ , then$a_{-m}=e^{i}\psi_{an-m}$ $(m=1,2, \cdots, n-1)$ .Theorem 3. ([1], [2]) The Toeplitz matrix $T$ on $\mathbb{C}^{n}$ is normal if andonly if, for some $\theta\in[0,2\pi)$ ,$\{$$a_{-m}=e^{i\theta}\overline{a_{m}}(m=1,2, \cdots, n-1)$ (type I)or $a_{-m}=e^{i\theta}a_{n-m}$ $(m=1,2, \cdots, n-1)$ (type II).Proof. By Theorem 2, we have only to prove that the equation (6) implies(type I) or (type II) because (6) is $.\mathrm{s}$atisfied clearly for each case of $a_{-k}’ \mathrm{s}$ suchas (type I) and (type II).(case 1) If $a_{j}=0(j=1,2, \cdots, n-1)$ , then $a_{-j}=0(j=1,2_{\text{ }}.\cdots, n-1)$ by(6-4) and hence (type I) and (type II) occur at the same time.(case 2) If there exists $j$ such that $a_{-j}=e^{i\varphi}\overline{a_{j}}(a_{j}\neq 0)$ and that $a_{n-j}=0$ ,then, by Lemma 4, (type I) occurs.(case 3) If there exists $j$ such that $a_{-j}=e^{i\psi}a_{n-j}(a_{n-j}\neq 0)$ and that $a_{j}=0$ ,then, by Lemma 5, (type II) occurs..’ If $a_{j}=0$ , then, by (6-3), $\overline{a-j}a-(n-j)=0$ and we have $a_{-j}=0$ or $a_{-(n-j)}=$$0$ . In the case where $a_{-j}\neq 0$ , by (6-4), $a_{-j}=e^{i\varphi,}\overline{aj}=0$ or $a_{-j}=e^{i\psi_{j}}a_{n-j}$ andwe have $a_{(n-j)}\neq 0$ and hence (case 3) occurs. In the case where $a_{-(?\iota^{-j)}}\neq 0$ , by(6-4), $a_{-}(n-j)=e^{i\varphi_{\mathcal{T}}}\iota-\mathcal{J}\overline{a_{n-}j}$ or $a_{-(n-j)}=e^{i\psi_{\mathit{7}\iota-j}}a_{j}=0$ and we have $a_{(_{7\iota-}j)}\neq 0$and hence (case 2) occurs. Therefore we may consider the case that $a_{-\dot{J}}=$$a_{-(n-j)}=0$ . And then, by (6), we have $\overline{a_{n-j}}a_{n-k}=0(k=1_{j}.2_{\vee}.\cdots, n-1)$and we may assume $a_{n-j}=0$ because, in the other case, (case 1) occurs. Andhenceforth we may assume that$a_{j}=0$ implies $a_{n-j}=0$ and then $a_{-j}=a_{-()}n-\dot{J}=0$ $(\#)$by (6-4). And hencethe equations of (type I) and (type II)are satisfied for each $j$ such as $a_{j}=0$ . (b)68And henceforth we may assume that there exists some $j_{0}$ such that $a_{j_{0}}\neq 0$and $a_{n-j_{0}}\neq 0$ . Then, by (6-4), (6-5) and (6-6), we have$d.$ ‘$\{$$a_{-j\mathrm{o}}=e^{i}\{i_{j}\mathrm{o}\overline{a\cdot}$$a_{-(n-j_{0})}=e^{i\varphi_{j}}J00\overline{a_{n-}j\mathrm{o}}$ or $\{$$a_{-j\mathrm{o}}=e^{i\psi_{j}}\mathrm{o}an-j0$$a_{-(}n-j_{0})=e^{i\psi_{j}}0aj_{0}$ .(case 4) If $a_{j}=0$ for all $j\neq j\mathrm{o}$ , then $\mathrm{c}\iota_{\iota-j},,=c\iota_{-j}=a_{-(?l}-j$ ) $=0$ for all $j\neq j_{0}$by $(\#)$ and (type I) or (type II) occurs by (b).(case 5) If there exists some $j_{1}$ such that $a_{j_{1}}\neq 0$ and that $| \frac{a_{\tau\iota-j_{0}}}{a_{\mathrm{j}_{0}}}|\neq|\frac{a_{n-j_{1}}}{a_{j_{1}}}|$ ,then, by Lemmas 1 and 2, (type I) or (type II) occurs because $a_{j_{1}}\neq 0$implies $a_{n-j_{1}}\neq 0$ by $(\#)$ .(case 6) If there exists some $j_{2}$ such that $a_{j_{2}}\neq 0,$ $| \frac{o_{n-j_{0}}}{a_{j_{0}}}|=|\frac{a_{n-\mathcal{J}2}}{a_{j_{2}}}|=|\frac{a_{j_{2}}}{a_{n-j_{2}}}|$and that $\frac{a_{n-j_{0}}}{\overline{a_{j_{0}}}}\neq\frac{a_{n-j_{2}}}{\overline{a_{j_{2}}}}$ , then, by Lemmas 3 and 2, (type I) or (typeII) occurs because $a_{j_{2}}\neq 0$ implies $a_{n-j_{2}}\neq 0$ by $(\#)$ .(case 7) If $| \frac{a_{n-J0}}{a_{j_{0}}}|=|\frac{a_{\tau\iota-j}}{a_{j}}|=|\frac{a_{j}}{a_{n-j}}|$ and $\frac{a_{n-j_{0}}}{\overline{a_{j_{0}}}}=\frac{a_{\tau\iota-j}}{\overline{a_{j}}}$ for all $j\neq j_{0}$ such as$a_{j}\neq 0$ , then, by Lemma 3 and by (b), (type I) and (type II) occur atthe same time because $a_{j}\neq 0$ implies $a_{n-j}\neq 0$ by $(\#)$ .References[1] Farenik, D. R., Krupnik, M., Krupnik, N. and Lee, W. Y., lVormal, Toeplitzmatrices, SIAM J. Matrix Anal. Appl., 17(1996), pp. 1037-1043.[2] Ito, T., Every nor$7r\iota al$, Toeplitz matri.’c is either of typeI or of type II,SIAM J. Matrix Anal. Appl., 17(1996), pp. 998-1006.69

Normal Toeplitz matrices on $\mathbb{C}^n$ (Operator Inequalities and Related Area)

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Normal Toeplitz matrices on $\mathbb{C}^n$ (Operator Inequalities and Related Area)

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Normal Toeplitz matrices on $\mathbb{C}^n$ (Operator Inequalities and Related Area)