%%%%% hhsw0306.tex %%%%% Version of 6 March 1996---HHSW (# 9117, # 126) %% Equ. (2.2) corrected, the Albrecht ref' completed, Aug '99 \magnification=\magstephalf \baselineskip = 15 pt \def\bz{{\bf z}} \def\B{\vrule width 0.07 in height 0.1 in depth 1 pt} \centerline{\bf APPROXIMABILITY BY WEIGHTED NORMS} \centerline{\bf OF THE STRUCTURED AND VOLUMETRIC SINGULAR VALUES} \centerline{\bf OF A CLASS OF NONNEGATIVE MATRICES} \bigskip \bigskip \centerline{by} \bigskip \bigskip \centerline{Daniel Hershkowitz \footnote{\dag}{The research of these authors was supported by their joint grant No. 90--00434 from the United States--Israel Binational Science Foundation, Jerusalem, Israel.} } \medskip {\it \centerline{Mathematics Department} \centerline{Technion -- Israel Institute of Technology} \centerline{Haifa 32000} \centerline{Israel}} \bigskip \medskip \centerline{Wenchao Huang \ddag} \medskip {\it \centerline{Mathematics Department} \centerline{University of Wisconsin--Madison} \centerline{Madison, Wisconsin 53706} \centerline{USA}} \bigskip \medskip \centerline{Hans Schneider \dag\footnote{\ddag}{The research of these authors was supported in part by NSF grant DMS--9123318 and DMS--9424346.}} \medskip {\it \centerline{Mathematics Department} \centerline{University of Wisconsin--Madison} \centerline{Madison, Wisconsin 53706} \centerline{USA}} \bigskip \medskip \centerline{Hans Weinberger} \medskip {\it \centerline{School of Mathematics} \centerline{University of Minnesota} \centerline{Vincent Hall, 206 Church St.} \centerline{Minneapolis, MN 55455--0487} \centerline{USA}} \vfill \centerline{6 March 1996} \break \bigskip \centerline{\bf APPROXIMABILITY BY WEIGHTED NORMS} \centerline{\bf OF THE STRUCTURED AND VOLUMETRIC SINGULAR VALUES} \centerline{\bf OF A CLASS OF NONNEGATIVE MATRICES} \bigskip \bigskip \bigskip \centerline{by} \bigskip \bigskip \bigskip \centerline{Daniel Hershkowitz} \centerline{Wenchao Huang} \centerline{Hans Schneider} \centerline{Hans Weinberger} \bigskip \bigskip \bigskip \bigskip \centerline{ABSTRACT} \bigskip A known result about the spectral radius of an irreducible nonnegative matrix is extended to all nonnegative matrices. By means of this result, it is shown that the structured singular value and the volumetric singular values of a class of nonnegative matrices can be approximated with arbitrary accuracy by the matrix norm induced by a weighted $\ell_2$ vector norm, and in the simplest case by a weighted $\ell_p$ vector norm for any $p$. %ACCEPTED \vfill \break %\bigskip \noindent{\bf 1. Introduction.} \medskip We begin with a survey of some results relating the spectral radius of a matrix to the norms of matrices similar to the given matrix. Let $A$ be a complex $n\times n$ matrix. If $||\ ||$ is any norm on the complex $n$-space, the corresponding induced matrix norm $||A||^0$ is defined as the supremum of the ratio $||A{\bf z}||/||{\bf z}||$ among complex nonzero vectors $\bf z$. Because the spectral radius $\rho(A)$ is defined as the largest absolute value of the eigenvalues of $A$, we obtain the well-known inequality $$ \rho(A)\le||A||^0 \leqno(1.1) $$ for any induced matrix norm, by choosing ${\bf z}$ to be an eigenvector. It was shown by A.~Householder ([H1, Theorem 4.4]), and the remark preceding it) that $\rho(A)$ is equal to the infimum of all induced matrix norms $||A||^0$. (See also [H2, p.46] or [HJ, Lemma 5.6.10].) The proof consists of observing that for every $\epsilon>0$ there is a nonsingular matrix $X_\epsilon$ such $X_\epsilon AX_\epsilon^{-1}$ is in Jordan form with the off-diagonal elements $\epsilon$. Householder then concludes that this implies that the matrix norm $||X_\epsilon AX_\epsilon^{-1}||_2^0$ induced by the $\ell_2$ norm is arbitrarily close to $\rho(A)$. It was shown by Bauer, Stoer, and Witzgall [BSW, Theorem 3], (see also [HJ, Theorem 5.6.37]) that the operator norm induced on a complex diagonal matrix by any absolute norm (that is, any norm whose value depends only on the absolute values of the components) is just the spectral radius of the matrix. Therefore the Householder argument shows that the matrix norm $||X_\epsilon AX_\epsilon^{-1}||^0$ induced by any absolute norm is arbitrarily close to $\rho(A)$. In particular, we see that for every $p\in[1,\infty]$ $$ \rho(A)=\inf_{X\ {\rm nonsingular}}||XAX^{-1}||_p^0, \leqno(1.2) $$ where $||\ ||_p^0$ denotes the matrix norm induced by the $\ell_p$ norm in the complex $n$-space $C^n$. See Friedland [F] for further discussion. For the special case of a real $n\times n$ matrix $A$ there exists a real similarity matrix such that $X_\epsilon AX_\epsilon^{-1}$ is in a block Jordan form whose diagonal blocks are $\lambda$ for real eigenvalues and $|\lambda|$ times a $2\times2$ orthogonal matrix for complex $\lambda$, and with the off diagonal blocks of order $\epsilon$. (See, e.g., [CL, p. 106, Prob. 40] and the proof given in [HJ, pp. 150--153].) The above argument shows that when $A$ is real and $p=2$, then (1.2) still holds if the infimum is taken only over the {\it real} nonsingular matrices. However, an example in the appendix shows that this extension is not true, even for $2\times 2$ matrices, when $p\ne2$. It was shown by Stoer and Witzgall [SW] that when $A$ is not only real but also entrywise positive, then for any $p\in[1,\infty]$ $$ \rho(A)=\min_{X\in{\cal X}}||XAX^{-1}||_p^0, \leqno(1.3) $$ where $$ {\cal X}=\{X:X\ {\rm real,\ diagonal,\ and\ positive\ definite}\}. \leqno(1.4) $$ This is a further improvement of (1.2) in two ways. The infimum is taken over the smaller set $\cal X$ of matrices $X$, and the infimum is attained. We observe that when $X\in{\cal X}$, the norm $||{\bf z}|| = ||X{\bf z}||_p$ which induces the matrix norm $||XAX^{-1}||_p^0$ is just a weighted $\ell_p$ norm. Recently, Albrecht [A] has generalized the Stoer-Witzgall result to irreducible nonnegative matrices. We now explain the contributions of this paper. In Section 2, by considering infima in place of minima, we extend the $\ell_2$ case of the Albrecht-Stoer-Witzgall theorem to arbitrary nonnegative matrices. In Section 3 we apply this result to produce certain classes of matrices for which the structured singular value introduced by Doyle [D] and the volumetric singular value introduced by Barmish and Polyak [BP] can be approximated with arbitrary accuracy by means of weighted $\ell_2$ norms. In Section 4 we show that in a special case the results in Section 3 can be extended to yield sharp weighted $\ell_p$ bounds for the structured and the volumetric singular values. \bigskip \noindent{\bf 2. Approximability Of The Spectral Radius.} \medskip If $A$ is a square matrix with nonnegative entries, it has a nonnegative eigenvalue, the so-called Perron eigenvalue, which is equal to its spectral radius $\rho(A)$, and there exists (at least) one nonnegative left eigenvector $\bf v$ and one nonnegative right eigenvector $\bf u$ corresponding to this eigenvalue. (See, e.g., ~[HJ, Theorem 8.3.1].) We call such eigenvectors {\bf Perron eigenvectors} of $A$. When $A$ is also irreducible, Frobenius showed that the left and right Perron eigenvectors are unique (up to a scalar multiples) and positive. (See, e.g., ~[HJ, Theorem 8.4.4].) \medskip The following simple proof of a generalization which covers irreducible matrices of the special $\ell_2$ case of the Stoer-Witzgall theorem is probably known, although we have been unable to find it written down. \medskip %CHANGES {\bf PROPOSITION 2.1}. {\it Let $A$ be a matrix with nonnegative entries and suppose that there exist left and right Perron eigenvectors $\bf u$ and $\bf v$ which are positive. If $$ X:={\rm diag}\{v_j^{1/2}u_j^{-1/2}\}, \leqno(2.1) $$ then $$ \rho(A)= ||XAX^{-1}||_2^0 \leqno(2.2) $$ so that the Stoer-Witzgall equation (1.3) is valid when $p=2$.} \medskip %END CHANGES {\it Proof.} Observe that, by (2.2), $X{\bf u}=X^{-1}{\bf v}$, so that $$ (XAX^{-1})^*(XAX^{-1})X{\bf u}=(X^{-1}A^*X)X\rho(A){\bf u}=\rho(A)^2X^{-1}{\bf v}=\rho(A)^2X{\bf u}. $$ Thus $\rho(A)^2$ is an eigenvalue of the nonnegative matrix $(XAX^{-1})^*(XAX^{-1})$, and the corresponding eigenvector $Xu$ is positive. Because a positive eigenvector of a nonnegative matrix must correspond to the spectral radius (see, e.g., ~[HJ, Corollary 8.1.30]), the spectral radius of this matrix is $\rho(A)^2$. Thus $||XAX^{-1}||_2^0=\rho(A)$, which is the statement of the Proposition. \hfill \B \bigskip We now extend the result in such a way that it also applies to all nonnegative matrices. \medskip {\bf THEOREM 2.2.} {\it Let $A$ be a matrix with nonnegative entries. Then $$ \rho(A)=\inf_{X\in{\cal X}}||XAX^{-1}||_2^0, \leqno(2.3) $$ where} $$ {\cal X}=\{X:X\ {\rm diagonal\ and\ positive\ definite}\}. $$ \medskip {\it Proof.} It was shown by Frobenius that there exists a permutation matrix $P$ such that the matrix $PAP^t$ is block upper triangular, with the diagonal blocks irreducible. We shall assume that this permutation has been done, so that the matrix $A$ has this form. By Proposition 2.1 there exists for each of the diagonal blocks $A_{jj}$ a positive definite diagonal matrix $X_j$ such that the matrix $$ D_j:=X_jA_{jj}X_j^{-1} $$ has the property $$ \rho(D_j)=||D_j||_2^0. \leqno(2.4) $$ For any positive $\epsilon$ we define the block diagonal matrix $$ X_\epsilon:={\rm diag}\{\epsilon^{-j} X_j\}. $$ It is easily verified that $$ X_\epsilon AX_\epsilon^{-1}=D+\epsilon E, \leqno(2.5) $$ where $D$ is the block diagonal matrix with the blocks $D_j$, and $E$ is a strictly upper triangular matrix whose entries are polynomials in $\epsilon$. %CHANGES Since $$ \rho(D)=\max_j\{\rho(D_j)\}, $$ and $$ ||D||_2^0=\max_j\{||D_j||_2^0\}, $$ the property (2.4) implies that $$ \rho(D)=||D||_2^0. \leqno(2.6) $$ %END CHANGES We see from this, (2.5), the continuity of the norm and the spectral radius, and the definition of $D$ that $$ \rho(A)=\rho(X_\epsilon AX_\epsilon^{-1})=\rho(D)+o(1) =||D||_2^0+o(1)=||X_\epsilon AX_\epsilon^{-1}||_2^0+o(1). $$ In other words, $$ \lim_{\epsilon\to 0}||X_\epsilon AX_\epsilon^{-1}||_2^0=\rho(A). $$ Since $X_\epsilon$ is in the class ${\cal X}$, this implies the statement (2.3) of the Theorem. \hfill \B \bigskip We note that a similar construction can be found in [S, p.~17] for a related problem. The example of the matrix $\pmatrix{1&1\cr0&1}$ shows that the infimum in (2.3) need not be attained. \bigskip \noindent{\bf 3. Structured And Volumetric Singular Values.} \medskip %CHANGES For any complex square matrix $A$ the matrix norm $||A||_2^0$ induced by the $\ell_2$ norm is equal to $\rho(A^*A)^{1/2}$. The latter quantity is also the largest singular value of $A$, and is often written as $\mu(A)$. By using the polar decomposition $A=VH$ where $V$ is unitary and $H$ is Hermitian positive semidefinite, one sees that $\mu(A)=\mu(H)=\rho(H)$. Since for any unitary matrix $U$ we have $\rho(UA)\le \mu(UA)=\mu(A)$ and since $\rho(V^*A)=\rho(H)=\mu(A)$, we see that $$ \mu(A)=\max_{U\in{\cal U}}\rho(UA), $$ where $\cal U$ is the group of unitary matrices. \medskip In order to study the robustness of feedback controls, Doyle [D] introduced the concept of {\it structured singular value}. Let the set of $n$ coordinate vectors of $C^n$ be partitioned into some number $\ell$ of disjoint subsets. The spans of the vectors in the subsets form $\ell$ orthogonal subspaces which together span $C^n$. %END CHANGES For the sake of simplicity, we suppose that the indices of the vectors in any one subset are contiguous. Then an $n\times n$ matrix is naturally partitioned into a block matrix in which each block represents a transformation from one of the prescribed subspaces into the same or a different subspace. For this reason the partition of $R^n$ into coordinate subspaces is called a {\it block structure}. We shall use $B$ to denote the block structure. Let ${\cal U}_B$ be the group of unitary matrices which are block diagonal in the given block structure $B$. Doyle's definition of the structured singular value $\mu_s(A)$ is\footnote{*}{ Note that $\mu_s$ depends upon the block structure $B$, so that it might have been better to denote it by $\mu_B(A)$.} $$ \mu_s(A):=\max_{U\in{\cal U}_B}\rho(UA). \leqno(3.1) $$ %END CHANGES We let ${\cal X}_B$ denote the set of all positive definite matrices which commute with all the matrices of ${\cal U}_B$. It is easily seen that $$ \eqalign{ {\cal X}_B=\{X:X\ &{\rm diagonal\ and\ positive\ definite,\ and\ the}\cr&{\rm diagonal \ entries\ of}\ X\ {\rm on\ each\ block\ are\ equal}\}.} \leqno(3.2) $$ %CHANGES In connection with the same problem, Barmish and Polyak [BP, p.~8] introduced the {\it volumetric singular value} which may be defined as $$ \mu_v(A):=\inf_{R\in {\cal X}_B\atop {\rm det}(R)=1}\mu_s(AR). \leqno(3.3) $$ The infimum in (3.3) need not be attained; see [BP, p.5]. %END CHANGES \medskip It was observed by Doyle [D] that because the matrices of ${\cal X}_B$ commute with those of ${\cal U}_B$, because a similarity transformation leaves the spectral radius invariant, and because a unitary matrix leaves the $\ell_2$ norm invariant, one finds that for any $U\in{\cal U}_B$ and any $X\in{\cal X}_B$ $$ \rho(UA)=\rho(XUAX^{-1})=\rho(UXAX^{-1})\le||XAX^{-1}||_2^0. \leqno(3.4) $$ By maximizing the left-hand side and minimizing the right, Doyle obtained the bound $$ \mu_s(A)\le\inf_{X\in{\cal X}_B}||XAX^{-1}||_2^0. \leqno(3.5) $$ By replacing $A$ by $AR$, defining $Y=RX^{-1}$, and minimizing both sides of the inequality (3.4), Barmish and Polyak obtained the bound $$ \mu_v(A)\le\inf_{X,Y\in{\cal X}_B\atop{\rm det}(XY)=1}||XAY^{-1}||_2^0. \leqno(3.6) $$ %CHANGES Because it is relatively easy to produce good algorithms to approximate the right-hand sides of (3.5) and (3.6), Doyle asked for conditions on $A$ which assure that equality holds in these bounds. We shall use the results of Section 2 to find such conditions. %END CHANGES \medskip Let ${\bf S}_B$ denote the $\ell$-dimensional subspace of $R^n$ which consists of those vectors whose components on each of the prescribed subspaces are equal. \medskip {\bf DEFINITION}: A matrix $A$ is said to be {\it adapted} to the block structure $B$ if ${\bf S}_B$ is an invariant subspace of both $A$ and its adjoint (complex transpose) $A^*$. \medskip It is easily seen that a matrix $A$ is adapted to the block structure $B$ if and only if each of the $\ell^2$ block matrices into which $A$ is partitioned by the block structure has constant row sums and constant column sums. \medskip We can now state our result on structured and volumetric singular values. \medskip {\bf THEOREM 3.1.} {\it Let $A$ be nonnegative and adapted to the block structure $B$. Then the structured singular value of $A$ is equal to the spectral radius of $A$, and equality holds in the inequalities (3.5) and (3.6).} \medskip %CHANGES {\it Proof.} We define the nonnegative $\ell\times\ell$ matrix $A_B$ whose $ij$ element is the common row sum of the $ij$ block of $A$. Suppose for the moment that $A_B$ is irreducible, and let ${\bf u}_B$ denote its (positive) right Perron eigenvector. Define the positive $n$-vector ${\bf u\in S}_B$ by requiring that its components in the $j$th block of $B$ equal the $j$th component of ${\bf u}_B$. The definitions of $A_B$, ${\bf u}_B$, and $\bf u$ show that $A{\bf u}=\rho(A_B){\bf u}$, so that $\bf u$ is a positive eigenvector of $A$. As in the proof of Proposition 2.1, this shows that $\rho(A)=\rho(A_B)$, so that $A$ has the positive right Perron eigenvector $\bf u$. Similar reasoning also shows that $A$ also has a positive left Perron eigenvector. One must work with the matrix $\tilde A_B$ whose $ij$ entry is the common column sum of the $ij$ block, and it easy to see that $\tilde A_B$ is irreducible if $A_B$ is irreducible. We have thus shown that if $A_B$ is irreducible, $A$ has positive left and right Perron eigenvectors in ${\bf S}_B$, even though $A$ itself may be reducible. Then Proposition 2.1 shows that if $X$ is defined by (2.2), $\rho(A)=||XAX^{-1}||_2^0$. If the matrix $A_B$ is reducible, we can put it into the Frobenius form by means of a permutation of its rows and columns. By performing the same permutations on the row and column blocks of $A$, we obtain a matrix which is adapted to a permutation of the partition $B$, and whose restriction to the permuted ${\bf S}_B$ is in Frobenius form. We suppose this permutation to have been done beforehand, so that the matrix $A_B$ is in the Frobenius form. Then the matrix $A$ is block upper triangular, with each block a union of the blocks of the structure $B$. The restriction of each of the diagonal blocks $A_{\alpha\alpha}$ to ${\bf S}_B$ is a diagonal block of the Frobenius matrix $A_B$, and is therefore irreducible. As we showed at the beginning of this proof, this property implies that there is a positive definite diagonal matrix $X^{(\alpha)}$ which is adapted to $B$ such that $||X^{(\alpha)}A_{\alpha\alpha}(X^{(\alpha)})^{-1}||_2^0=\rho(A_{\alpha\alpha})$. As in the proof of Theorem 2.2, we now define $$ X_\epsilon={\rm diag}\{\epsilon^{-\alpha}X^{(\alpha)}\}, $$ and show that $$ \rho(A)=\lim_{\epsilon\to0}||X_\epsilon A X_\epsilon^{-1}||_2^0. $$ Since $X_\epsilon\in {\cal X}_B$, the right-hand side is bounded below by the right-hand side of (3.5). By definition, the left-hand side is bounded above by $\mu_s(A)$ which is the left-hand side of (3.5). We conclude that $\mu_s(A)=\rho(A)$, and that the two sides of (3.5) are equal. In order to obtain equality in (3.6), we see from the definition (3.3) that there is an $R_\epsilon\in{\cal X}_B$ with determinant 1 which makes $\mu_s(AR_\epsilon)$ close to $\mu_v(A)$. We use the above construction with $A$ replaced by $AR_\epsilon$ to find an $X_\epsilon\in {\cal X}_B$ so that $||X_\epsilon AR_\epsilon X_\epsilon^{-1}||_2^0$ is close to $\mu_s(AR_\epsilon)$ and hence close to $\mu_v(A)$. Equality in (3.6) follows from defining $Y_\epsilon=R_\epsilon X_\epsilon^{-1}$, and the Theorem is proved. \hfill \B \bigskip REMARK 1: The above proof shows that if, in addition to satisfying the hypotheses of Theorem 3.1, the matrix $A$ is block irreducible in the sense that there is no nontrivial direct sum of subspaces of the partition $B$ which is an invariant subspace of $A$, then there is a matrix $X\in{\cal X}_B$ for which $||XAX^{-1}||_2^0=\rho(A)$. \medskip REMARK 2: Doyle [D] also considered a definition of structured singular value in which the group ${\cal U}_B$ in the definition (3.1) is replaced by its subgroup of real orthogonal matrices. Because Theorem 3.1 shows that $\mu_s(A)=\rho(A)$, the maximum in (3.1) is attained when $U=I$, which is real and orthogonal. Thus the statement of Theorem 3.1 is still valid for this definition. \medskip We now observe that if $A$ is any matrix and $V$ and $W$ are any matrices in ${\cal U}_B$, then replacing $A$ by $VAW$ does not change either side of (3.5) or (3.6). We can thus immediately generalize the class of matrices for which equality in (3.5) and (3.6) holds. \medskip {\bf THEOREM 3.2.} {\it If $A$ is any complex matrix with the property that there exist matrices $V$ and $W$ in ${\cal U}_B$ such that $VAW$ is nonnegative and adapted to the block structure $B$, then equality holds in the bounds (3.5) and (3.6).} %END CHANGES \hfill \B \bigskip \noindent{\bf 4. Some Results In $\ell_p$.} \medskip Additional results can be obtained for the particular block structure $B_1$ in which all the blocks are $1\times 1$. In this case the set ${\cal U}_{B_1}$ consists of the diagonal unitary matrices, and these preserve not only the $\ell_2$ norm, but all the $\ell_p$ norms. Therefore, we can immediately replace $\ell_2$ by any $\ell_p$ in the bounds (3.5) and (3.6): $$ \mu_s(A)\le\inf_{X\in{\cal X}_{B_1}}||XAX^{-1}||_p^0 \leqno(4.1) $$ $$ \mu_v(A)\le\inf_{X,Y\in{\cal X}_{B_1}\atop{\rm det}(XY)=1}||XAY^{-1}||_p^0. \leqno(4.2) $$ We note that the set ${\cal X}_{B_1}$ is just the set ${\cal X}$ of all diagonal positive definite matrices, and that any matrix is adapted to the block structure $B_1$. Albrecht [A] has recently obtained the following extension to nonnegative irreducible matrices of the result of Stoer and Witzgall [SW] for positive matrices. This analogue of Proposition 2.1 will permit us to extend Theorem 3.1 to this case. \bigskip {\bf PROPOSITION 4.1.} {\it Let $A$ be an irreducible nonnegative matrix with the left and right Perron eigenvectors $\bf v$ and $\bf u$. For any $1\le p\le\infty$ the matrix $$ X={\rm diag}(v_j^{1/p}u_j^{(1-p)/p}) \leqno(4.3) $$ has the property that $$ \eqalignno{ & \rho(A)=||XAX^{-1}||_p^0. & \B \cr } $$ } By replacing Proposition 2.1 by Proposition 4.1 in the proof of Theorem 2.2, we find the following extension. \medskip %CHANGES {\bf THEOREM 4.2.} {\it If $A$ is nonnegative and $1\le p\le\infty$, then $$ \eqalignno{ & \rho(A)=\inf_{X\in{\cal X}}||XAX^{-1}||_p^0. & \B \cr } $$ } Question: For which induced matrix norms does Theorem 4.2 hold? %END CHANGES \medskip By replacing Theorem 2.2 by Theorem 4.2 in the proof of Theorems 3.1 and Corollary 3.2, we immediately find the following result. \medskip %CHANGES {\bf THEOREM 4.3.} {\it Let $B_1$ be the block structure which consists entirely of $1\times1$ blocks. If the complex matrix $A$ has the property that there are diagonal unitary matrices $V$ and $W$ such that the matrix $VAW$ is nonnegative, then for any $p\in[1,\infty]$ $$ \mu_s(A)=\inf_{X\in{\cal X}}||XAX^{-1}||_p^0=\rho(VAW), $$ and $$ \eqalignno{ & \mu_v(A)=\inf_{X,Y\in{\cal X}\atop {\rm det}(XY)=1}||XAY||_p^0. & \B \cr } $$ } REMARK 3. If $C$ is a complex matrix, an inequality which goes back to Frobenius asserts that $\rho(C)\le\rho(|C|)$, where $|C|$ is the matrix whose entries are the absolute values of the corresponding entries of $C$ (see, e.g., [HJ, Theorem 8.1.18]). This inequality applied to (3.1) yields an alternative proof of the equality $\mu_s(A)=\rho(VAW)$ in Theorem 4.3. \bigskip \noindent{\bf Appendix: A counterexample} \medskip We shall show that when $p\ne2$ and $A$ is the $2\times2$ matrix $\pmatrix{1&-1\cr1&1}$, equality does not hold in (1.2) when the infimum is taken only over real matrices $X$. Any real matrix which is similar to $A$ must have the same trace and determinant. Therefore if $X$ is real, $XAX^{-1}$ has the form $$ XAX^{-1}=\pmatrix{1+\alpha&-{1+\alpha^2\over c}\cr c&1-\alpha} $$ with $\alpha$ and $c$ real. By inserting the two coordinate vectors into the definition of the induced matrix norm, we find that $||XAX^{-1}||_p^0$ is bounded below by the larger of the $\ell_p$ norms of the two columns. Because the first of these norms is increasing in $|c|$ while the second is decreasing, one finds the smallest such lower bound with respect to the choice of $c$ by equating the two column norms. This leads to the lower bound $$ \{||XAX^{-1}||_p^0\}^p\ge{1\over2}\{|1+\alpha|^p+|1-\alpha|^p +\sqrt{(|1+\alpha|^p+|1-\alpha|^p)^2+4(|1+\alpha^2|^p-|1-\alpha^2|^p)}\}. $$ This expression has its minimum at $\alpha=0$, so that $||XAX^{-1}||_p^0\ge2^{1/p}$ for $1\le p<\infty$. It is easily verified that for the above matrix $A$, $\|A\|_1^0=2$ and $\|A\|_2^0=2^{1/2}$. Then for $12$, we use the fact that the matrix norm induced on a matrix by $\ell_p$ is equal to the norm induced on its conjugate transpose by the conjugate norm $\ell_q$ to conclude that $\inf||XAX^{-1}||_p^0$ $=\max\{2^{1/p},2^{(p-1)/p}\}$ for every $p$. Since $\rho(A)=2^{1/2}$, we conclude that when $p\ne2$, the right-hand side of (1.2) with $X$ restricted to real matrices is strictly larger than the left-hand side. \medskip {\it Acknowledgment}: We thank Michael Neumann for comments which have helped to improve this paper. \bigskip \bigskip %\vfill {} %\break \noindent{\bf BIBLIOGRAPHY} \medskip \item{A}J. Albrecht, {\it Minimal Norms of Non-Negative, Irreducible Matrices}, Lin. Alg. Appl. 249:255-258 (1996) \item{BP}B.~R.~Barmish and B.~T.~Polyak, {\it The Volumetric Singular Value and Robustness of Feedback Control Systems}, to appear in IEEE Trans. Aut. Cont. \item{BSW}F.~L.~Bauer, J.~Stoer and C.~Witzgall, {\it Absolute and monotonic norms}, Num. 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