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A unitary with a large fixed space

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The proof of the planar boundary theorem needs a unitary matrix that mixes \(k\) copies of a word disk and fixes many vectors. This lesson proves the statement that supplies it [OpenAI-K, Lemma 3.1]: if signs \(s_1,\ldots,s_n\) add up to \(\pm1\), then every unitary \(P\) of size \(nk\) can be multiplied by a block-diagonal matrix \(\operatorname{diag}(X^{s_1},\ldots,X^{s_n})\), with one unitary \(X\) of size \(k\), so that the product fixes a \(k\)-dimensional subspace. The proof is a degree computation modulo two, in the spirit of Gerstenhaber and Rothaus [Gerstenhaber–Rothaus 1962, Theorem 1]: a map between manifolds of the same dimension has odd degree, so it is onto. To make the dimensions agree, the domain records a unitary together with a \(k\)-plane that it fixes.

We use degree modulo two, Definition 4.1, the homotopy invariance stated after it, Lemma 4.2 and Corollary 4.3 of Regular values and degree modulo two, and the inverse function theorem of the core course Real Analysis II [Lebl, Theorem 8.5.1].

Throughout, matrices carry the operator norm, \(\mathrm U(m)\) is the group of unitary \(m\times m\) matrices, and \(\mathfrak u(m)\) is the real vector space of skew-Hermitian \(m\times m\) matrices, of real dimension \(m^2\). The exponential \(e^S=\sum_{n\ge0}S^n/n!\) of \(S\in\mathfrak u(m)\) is unitary, since \((e^S)^*=e^{S^*}=e^{-S}\).

1. Unitary groups and graph charts

Lemma 1.1 (unitary groups). For \(\|T\|<1\) put \(\log(I+T)=\sum_{n\ge1}(-1)^{n+1}T^n/n\).

  1. If \(\|S\|<\log2\), then \(\|e^S-I\|<1\) and \(\log e^S=S\). If \(\|T\|<1\), then \(e^{\log(I+T)}=I+T\).
  2. If \(W\in\mathrm U(m)\) and \(\|W-I\|<1\), then \(\log W\in\mathfrak u(m)\).
  3. For \(W_0\in\mathrm U(m)\), the maps \(S\mapsto W_0e^S\) on \(\{S\in\mathfrak u(m):\|S\|<\log2\}\) are homeomorphisms onto open subsets of \(\mathrm U(m)\) whose transition maps are smooth. So \(\mathrm U(m)\) is a compact smooth manifold of dimension \(m^2\). It is connected.

Proof. (1) The power series identities \(\log(e^x)=x\) and \(e^{\log(1+y)}=1+y\) hold for real or complex numbers with \(|x|<\log2\), \(|y|<1\), and the series converge absolutely. Substituting a matrix into absolutely convergent power series in one variable respects composition, so the same identities hold for \(S\) and \(T\); and \(\|e^S-I\|\le e^{\|S\|}-1<1\).

(2) \(W\) is normal, so \(W=V\operatorname{diag}(\lambda_j)V^*\) with \(V\) unitary and \(|\lambda_j|=1\), and \(\|W-I\|=\max_j|\lambda_j-1|<1\). Then \(\log W=V\operatorname{diag}(\log\lambda_j)V^*\), where \(\log\lambda_j\) is the sum of the scalar series. By (1), \(e^{\log\lambda_j}=\lambda_j\) has modulus one, so \(\log\lambda_j\) is purely imaginary, and \(\log W\) is skew-Hermitian.

(3) \(\mathrm U(m)\) is closed and bounded in the space of matrices, hence compact. The map \(S\mapsto W_0e^S\) is continuous and injective on the ball of radius \(\log2\), with continuous inverse \(W\mapsto\log(W_0^{-1}W)\) on the open set of those \(W\) with \(\|W_0^{-1}W-I\|<1\) and \(\|\log(W_0^{-1}W)\|<\log2\), by (1) and (2). The images cover \(\mathrm U(m)\), and the transition maps \(S\mapsto\log(W_1^{-1}W_0e^S)\) are given by convergent power series, hence smooth. Every unitary is \(V\operatorname{diag}(e^{i\theta_j})V^*\) with \(V\) unitary, and \(t\mapsto V\operatorname{diag}(e^{it\theta_j})V^*\) joins it to \(I\); so \(\mathrm U(m)\) is connected. \(\square\)

Grassmannians. Let \(\operatorname{Gr}_k(\mathbb C^m)\) be the set of complex \(k\)-dimensional subspaces of \(\mathbb C^m\). We identify a subspace \(E\) with its orthogonal projection \(p_E\) and give the Grassmannian the topology of these matrices. It is compact, being the image of the compact group \(\mathrm U(m)\) under \(g\mapsto gp_{E_0}g^*\) for a fixed \(E_0\): every \(k\)-plane is \(gE_0\) for a unitary \(g\) that carries an orthonormal basis of \(E_0\), extended to one of \(\mathbb C^m\), to one of \(E\).

Fix \(E_0\in\operatorname{Gr}_k(\mathbb C^m)\), put \(r=m-k\), and write vectors of \(\mathbb C^m=E_0\oplus E_0^\perp\) as pairs \((v,y)\), and matrices in block form. For a complex \(r\times k\) matrix \(Z\), viewed as a linear map \(E_0\to E_0^\perp\), put \[ K_Z=\begin{pmatrix}0&-Z^*\\Z&0\end{pmatrix}\in\mathfrak u(m). \]

Lemma 1.2 (graph charts).

  1. If \(\|p_E-p_{E_0}\|<1\), then \(E\) is the graph \(\{(v,Lv):v\in E_0\}\) of exactly one linear map \(L=L(E):E_0\to E_0^\perp\), and \(L(E)\) depends smoothly on \(p_E\).
  2. If \(\|Z\|<\pi/2\), then \(e^{K_Z}E_0\) is the graph of \(\Lambda(Z)=Zf(Z^*Z)\), where \(f(x)=\tan(\sqrt x)/\sqrt x=\sum_{j\ge0}c_jx^j\) for \(0\le x<\pi^2/4\), with \(f(0)=1\). The map \(\Lambda\) is smooth, and its derivative at \(0\) is the identity, so it maps a neighbourhood of \(0\) diffeomorphically onto a neighbourhood of \(0\).

Proof. (1) Write \(p_0=p_{E_0}\) and \(T=p_0p_Ep_0\) on \(E_0\). For \(v\in E_0\), \(\|Tv-v\|=\|p_0(p_E-p_0)v\|<\|v\|\) if \(v\ne0\), so \(T\) is invertible on \(E_0\), with inverse depending smoothly on \(p_E\) (Neumann series). For \(v\in E_0\) the vector \(u=p_ET^{-1}v\) lies in \(E\) and \(p_0u=TT^{-1}v=v\), so \(u=(v,Lv)\) with \(L=(I-p_0)p_ET^{-1}\). The graph of \(L\) is a \(k\)-dimensional subspace of \(E\), so it is \(E\). If \(E\) were also the graph of \(L'\), then \((v,Lv)-(v,L'v)\in E\) would lie in \(E_0^\perp\); but no nonzero \(u\in E\) has \(p_0u=0\), since then \(\|u\|=\|(p_E-p_0)u\|<\|u\|\). So \(L'=L\).

(2) From \(K_Z^2=\operatorname{diag}(-Z^*Z,-ZZ^*)\), the even and odd terms of the exponential series give \[ \begin{gathered} e^{K_Z}\begin{pmatrix}v\\0\end{pmatrix}=\begin{pmatrix}c(Z^*Z)v\\Zs(Z^*Z)v\end{pmatrix},\\ c(x)=\sum_{j\ge0}\frac{(-x)^j}{(2j)!},\\ s(x)=\sum_{j\ge0}\frac{(-x)^j}{(2j+1)!}, \end{gathered} \] so that \(c(x)=\cos\sqrt x\) and \(s(x)=\sin\sqrt x/\sqrt x\) for \(x\ge0\). The eigenvalues of \(Z^*Z\) are squares of numbers in \([0,\|Z\|]\subseteq[0,\pi/2)\), so \(c(Z^*Z)\) is invertible, and \(e^{K_Z}E_0\) is the graph of \(Zs(Z^*Z)c(Z^*Z)^{-1}=Zf(Z^*Z)\). The series \(\sum_jc_jZ(Z^*Z)^j\) converges for \(\|Z\|<\pi/2\), so \(\Lambda\) is smooth; \(\Lambda(Z)=Z+O(\|Z\|^3)\), so its derivative at \(0\) is the identity, and the inverse function theorem applies. \(\square\)

2. The incidence manifold

Lemma 2.1 (the incidence manifold). Let \(1\le k\le m\) and \(r=m-k\). The set \[ \begin{gathered} \mathcal R=\{(W,E)\in\mathrm U(m)\times\operatorname{Gr}_k(\mathbb C^m):\\ W|_E=\mathrm{id}_E\} \end{gathered} \] is a compact smooth manifold of dimension \(m^2-k^2=2kr+r^2\). Charts are obtained as follows. For \((W_1,E_1)\in\mathcal R\) there are \(g\in\mathrm U(m)\) with \(gE_0=E_1\), and \(Y_1\in\mathrm U(r)\) with \(g^{-1}W_1g=I_k\oplus Y_1\); then \[ \begin{gathered} \chi(Z,B)=\Bigl(g\,e^{K_Z}\bigl(I_k\oplus Y_1e^B\bigr)e^{-K_Z}g^{-1},\\ g\,e^{K_Z}E_0\Bigr), \end{gathered} \] for complex \(r\times k\) matrices \(Z\) and \(B\in\mathfrak u(r)\), both small, maps a neighbourhood of \(0\) homeomorphically onto a neighbourhood of \((W_1,E_1)\) in \(\mathcal R\).

Proof. \(\mathcal R\) is defined by the closed condition \((W-I)p_E=0\) in a compact space, so it is compact. Given \((W_1,E_1)\), choose \(g\) as in the description of the Grassmannian. Then \(g^{-1}W_1g\) fixes \(E_0\) pointwise; a unitary that preserves \(E_0\) preserves \(E_0^\perp\), so \(g^{-1}W_1g=I_k\oplus Y_1\) with \(Y_1\in\mathrm U(r)\).

The point \(\chi(Z,B)\) lies in \(\mathcal R\): its unitary fixes \(ge^{K_Z}E_0\) pointwise. Conversely, we construct a continuous inverse. Let \((W,E)\in\mathcal R\) be close to \((W_1,E_1)\). Then \(g^{-1}E\) is close to \(E_0\), so it is the graph of \(L(g^{-1}E)\) (Lemma 1.2(1)), and there is exactly one small \(Z\) with \(\Lambda(Z)=L(g^{-1}E)\), that is \(e^{K_Z}E_0=g^{-1}E\) (Lemma 1.2(2)); it depends continuously, indeed smoothly, on \(p_E\). The unitary \(e^{-K_Z}g^{-1}Wge^{K_Z}\) fixes \(E_0\) pointwise, so it is \(I_k\oplus Y\) with \(Y\in\mathrm U(r)\) close to \(Y_1\), and \(B=\log(Y_1^{-1}Y)\) (Lemma 1.1). The map \((W,E)\mapsto(Z,B)\) so defined is continuous, indeed given by smooth formulas in the entries of \(W\) and \(p_E\), and sends \((W_1,E_1)\) to \((0,0)\). It inverts \(\chi\) on both sides near these points: for small \((Z,B)\), the construction applied to \(\chi(Z,B)\) returns \(Z\), by the uniqueness in Lemma 1.2(2), and then \(Y=Y_1e^B\) and \(B\), by Lemma 1.1(1); and every \((W,E)\in\mathcal R\) close to \((W_1,E_1)\) equals \(\chi\) of its small parameters. So \(\chi\) maps a neighbourhood of \(0\) homeomorphically onto a neighbourhood of \((W_1,E_1)\) in \(\mathcal R\). A chart composed with the inverse of another is smooth, because \(\chi\) is smooth as a map into pairs of matrices \((W,p_E)\), with \(p_E=ge^{K_Z}p_{E_0}e^{-K_Z}g^{-1}\), and the inverse of a chart is given by smooth formulas in \((W,p_E)\). The parameters \(Z\) and \(B\) have \(2kr\) and \(r^2\) real coordinates. If \(r=0\), \(\mathcal R\) is the single point \((I,\mathbb C^m)\), of dimension \(0=m^2-k^2\). \(\square\)

3. A homotopy to a model

Fix signs \(s_1,\ldots,s_n\in\{1,-1\}\) with \(\varepsilon=\sum_is_i\in\{1,-1\}\), an integer \(k\ge1\), and \(m=nk\). Split \(\mathbb C^m=E_1\oplus\cdots\oplus E_n\) into coordinate blocks of dimension \(k\), and for \(X\in\mathrm U(k)\) and \(P\in\mathrm U(m)\) put \[ F(X)=\operatorname{diag}(X^{s_1},\ldots,X^{s_n})\,P . \] Since \(\varepsilon\ne0\), some index \(i_0\) has \(s_{i_0}=\varepsilon\). Put \(E_0=E_{i_0}\), \(r=m-k\), and \[ \begin{gathered} F_0(X)=X^{\varepsilon}\oplus(-I_r)\\ \text{on }E_0\oplus E_0^\perp . \end{gathered} \]

Lemma 3.1. There is a continuous map \(\Phi:[0,1]\times\mathrm U(k)\to\mathrm U(m)\) with \(\Phi(0,\cdot)=F\) and \(\Phi(1,\cdot)=F_0\).

Proof. We concatenate three homotopies; each is continuous in both variables.

Remove \(P\). Write \(P=V\operatorname{diag}(e^{i\theta_j})V^*\) with \(V\) unitary and real \(\theta_j\), and replace \(P\) by \(V\operatorname{diag}(e^{i(1-u)\theta_j})V^*\), \(0\le u\le1\).

Cancel pairs. The numbers \(n_\pm\) of signs \(\pm1\) satisfy \(n_+-n_-=\varepsilon\), so the indices other than \(i_0\) split into pairs \(\{i,j\}\) with \(s_i=1\) and \(s_j=-1\). On \(E_i\oplus E_j\) put \[ \begin{gathered} R_u=\begin{pmatrix}\cos(\pi u/2)I_k&-\sin(\pi u/2)I_k\\ \sin(\pi u/2)I_k&\cos(\pi u/2)I_k\end{pmatrix},\\ H_u(X)=\operatorname{diag}(X,I_k)\,R_u\\ \cdot\operatorname{diag}(I_k,X^{-1})\,R_u^{-1}. \end{gathered} \] Then \(H_0(X)=\operatorname{diag}(X,X^{-1})\), and since \(R_1\operatorname{diag}(I_k,X^{-1})R_1^{-1}=\operatorname{diag}(X^{-1},I_k)\), \(H_1(X)=I\). Apply \(H_u\) on all pairs at once, keeping \(X^\varepsilon\) on \(E_0\).

Move the identity to \(-I\). On \(E_0^\perp\) replace \(I\) by \(e^{i\pi u}I\), \(0\le u\le1\). \(\square\)

4. The unique preimage of the identity

Let \(\mathcal M=\mathrm U(k)\times\mathcal R\), with \(\mathcal R\) as in Lemma 2.1; it is a compact smooth manifold of dimension \(k^2+(m^2-k^2)=m^2\), the dimension of \(\mathrm U(m)\). For a map \(\Psi:\mathrm U(k)\to\mathrm U(m)\) define \[ \begin{gathered} G_\Psi:\mathcal M\longrightarrow\mathrm U(m),\\ G_\Psi(X,W,E)=\Psi(X)\,W^{-1} . \end{gathered} \] A point with \(G_\Psi(X,W,E)=I\) has \(\Psi(X)=W\), so \(\Psi(X)\) fixes the \(k\)-plane \(E\) pointwise.

Lemma 4.1. Let \(W_0=I_k\oplus(-I_r)\).

  1. The only point of \(\mathcal M\) mapped to \(I\) by \(G_{F_0}\) is \((I_k,W_0,E_0)\).
  2. In the chart \((A,Z,B)\mapsto(X,W,E)\) of \(\mathcal M\) at this point given by \(X=e^A\), \(W=e^{K_Z}(I_k\oplus(-e^B))e^{-K_Z}\) and \(E=e^{K_Z}E_0\), with \(A\in\mathfrak u(k)\) (Lemma 1.1 and Lemma 2.1 with \(g=I\), \(Y_1=-I_r\)), and the chart \(\log\) of \(\mathrm U(m)\) at \(I\), the map \(G_{F_0}\) is smooth near \(0\), and its derivative at \(0\) is \[ (A,Z,B)\longmapsto\begin{pmatrix}\varepsilon A&2Z^*\\-2Z&-B\end{pmatrix}, \] a linear isomorphism of \(\mathfrak u(k)\times M_{r\times k}(\mathbb C)\times\mathfrak u(r)\) onto \(\mathfrak u(m)\).

Proof. (1) If \(G_{F_0}(X,W,E)=I\), then \(W=F_0(X)=X^\varepsilon\oplus(-I_r)\). Every fixed vector of this matrix lies in \(E_0\), and \(W\) fixes the \(k\)-dimensional space \(E\) pointwise, so \(E=E_0\) and \(X^\varepsilon=I_k\). Since \(\varepsilon=\pm1\), \(X=I_k\) and \(W=W_0\). Conversely this point is mapped to \(I\).

(2) The chart is the product of the chart of Lemma 1.1 at \(I_k\) and the chart of Lemma 2.1, and \(G_{F_0}\) is a composite of products, inverses, exponentials and \(\log\) near \(I\), so it is smooth. Differentiate along \(t\mapsto(tA,tZ,tB)\) at \(t=0\). The derivative of \(F_0(e^{tA})=e^{t\varepsilon A}\oplus(-I_r)\) is \(\dot F=\operatorname{diag}(\varepsilon A,0)\). The derivative of \(W(t)=e^{tK_Z}(I_k\oplus(-e^{tB}))e^{-tK_Z}\) is \[ \begin{aligned} \dot W&=K_ZW_0-W_0K_Z+\operatorname{diag}(0,-B)\\ &=\begin{pmatrix}0&2Z^*\\2Z&-B\end{pmatrix}. \end{aligned} \] At \(t=0\), \(F_0=W_0=W_0^{-1}\), so the derivative of \(F_0W^{-1}\) is \(\dot FW_0^{-1}-F_0W_0^{-1}\dot WW_0^{-1}\), that is \(\dot FW_0-\dot WW_0\), which is the displayed matrix; the derivative of \(\log\) at \(I\) is the identity. The linear map is injective, and both spaces have real dimension \(k^2+2kr+r^2=m^2\). If \(r=0\), the map is \(A\mapsto\varepsilon A\). \(\square\)

5. The fixed-space lemma

Lemma 5.1 (a unitary with a large fixed space; [OpenAI-K, Lemma 3.1]). Let \(s_1,\ldots,s_n\in\{1,-1\}\) with \(\sum_is_i\in\{1,-1\}\), let \(k\ge1\) and \(m=nk\). For every \(P\in\mathrm U(m)\) there is \(X\in\mathrm U(k)\) such that \[ F(X)=\operatorname{diag}(X^{s_1},\ldots,X^{s_n})\,P \] fixes pointwise a complex subspace of dimension at least \(k\).

Proof. \(\mathcal M\) is a closed smooth manifold of dimension \(m^2\) (Lemma 2.1), and \(\mathrm U(m)\) is a connected closed smooth manifold of the same dimension (Lemma 1.1). By Lemma 4.1 above and Lemma 4.2 of the degree lesson, \(\deg_2G_{F_0}=1\): the fibre over \(I\) is one point, at which \(G_{F_0}\) is differentiable in charts with invertible derivative. The homotopy \(\Phi\) of Lemma 3.1 gives the homotopy \((u,X,W,E)\mapsto\Phi(u,X)W^{-1}\) from \(G_F\) to \(G_{F_0}\), so \(\deg_2G_F=1\) by homotopy invariance. By Corollary 4.3 of the degree lesson, \(G_F\) is onto; a point with \(G_F(X,W,E)=I\) gives \(F(X)=W\), which fixes the \(k\)-plane \(E\) pointwise. \(\square\)

The conclusion holds for an arbitrary fixed \(P\); no continuity in \(P\) is claimed. The hypothesis \(\sum_is_i=\pm1\) enters in two places: it lets all blocks but one cancel in pairs, and it makes \(X^\varepsilon=I\) force \(X=I\), so that the preimage is a single point. For \(|\sum_is_i|\ge2\) the solutions of \(X^\varepsilon=I\) form a set of positive dimension, and the companion manuscript on nonsingular systems replaces the count of a single preimage by a computation in cohomology [OpenAI-H, Section 2]; the lesson Simultaneous fixed spaces proves that version, for several unitary variables. Without any condition the statement fails (Exercise 6.2).

6. Exercises

6.1 (easy). Prove Lemma 5.1 directly when \(n=1\).

6.2 (medium). Let \(n=2\), \(s=(1,-1)\), \(k=1\) and \(P=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\). Show that \(\operatorname{diag}(x,x^{-1})P\) has no eigenvalue \(1\) for any \(x\in\mathrm U(1)\).

6.3 (easy). Let \(m=2\), \(k=1\). Describe \(\mathcal R\) and check that its dimension is \(3\).

6.4 (easy). Prove Lemma 5.1 directly when \(P\) is block-diagonal, \(P=\operatorname{diag}(P_1,\ldots,P_n)\) with \(P_i\in\mathrm U(k)\).

6.5 (medium). For \(k=r=1\) and \(Z=z\in\mathbb C\), compute \(e^{K_z}\) and check Lemma 1.2(2).

7. Solutions

6.1. Then \(s_1=\varepsilon\) and \(F(X)=X^\varepsilon P\). For \(X=P^{-\varepsilon}\), \(X^\varepsilon=P^{-\varepsilon^2}=P^{-1}\), so \(F(X)=I\), which fixes all of \(\mathbb C^k\).

6.2. \(F(x)=\begin{pmatrix}0&x\\-x^{-1}&0\end{pmatrix}\) has trace \(0\) and determinant \(1\), so its characteristic polynomial is \(\lambda^2+1\), with roots \(\pm i\).

6.3. A point is a pair of a unitary \(W\) and a line \(E\) that it fixes; \(W\) acts on \(E^\perp\) by a number of modulus one. So \(\mathcal R\) is the set of pairs (line in \(\mathbb C^2\), number of modulus one), the product \(\mathbb{CP}^1\times\mathrm U(1)\), of dimension \(2+1=3=4-1\).

6.4. Then \(F(X)=\operatorname{diag}(X^{s_1}P_1,\ldots,X^{s_n}P_n)\). Let \(s_{i_0}=\varepsilon\) and \(X=P_{i_0}^{-\varepsilon}\); the block \(X^{\varepsilon}P_{i_0}=I_k\), so \(F(X)\) fixes \(E_{i_0}\) pointwise.

6.5. \(K_z^2=-|z|^2I_2\), so \(e^{K_z}=\begin{pmatrix}\cos|z|&-\bar z\,\frac{\sin|z|}{|z|}\\ z\,\frac{\sin|z|}{|z|}&\cos|z|\end{pmatrix}\) (with \(\frac{\sin|z|}{|z|}=1\) at \(z=0\)). It maps \((1,0)\) to \((\cos|z|,\ z\sin|z|/|z|)\), which spans the graph of multiplication by \(z\tan|z|/|z|=zf(|z|^2)\).

References