Is this condition on the eigenvalues of a matrix rigid under orthogonal multiplication?

Let $$\Sigma=\text{diag}(-\sigma_1,\sigma_2,\dots,\sigma_n)$$ be a diagonal matrix, where $$0\le\sigma_1 < \sigma_2 \le \sigma_3 \le \dots \le \sigma_n$$.

Note that I assume a strict inequality between $$\sigma_1$$ and $$\sigma_2$$.

Now, let $$Q \in \text{SO}(n)$$, and suppose that $$Q^T\Sigma$$ is symmetric, and that it has at most one negative eigenvalue, and the absolute value of the negative eigenvalue is less than or equal to the next-smallest eigenvalue.

Is it true that $$Q=\text{Id}$$?

To state it differently-we have here a diagonal matrix $$\Sigma$$ satisfying some condition on its eigenvalues. We then multiply it by an orthogonal matrix $$Q^T$$, and we require the condition to be preserved. Does this force $$Q=\text{Id}$$?

Edit:

I prove below that the eigenvalues of $$Q^T\Sigma$$ are the same as those of $$\Sigma$$. Does it help showing that $$Q=\text{Id}$$?

Indeed, since we assumed that $$Q^T\Sigma$$ is symmetric, its singular values $$\sigma_i$$ are the absolute value of its eigenvalues $$\lambda_i$$, i.e.

$$|\lambda_i(Q^T\Sigma)|=\sigma_i(Q^T\Sigma)=\sigma_i(\Sigma)=\{ \sigma_1,\dots,\sigma_n\},$$

so $$\lambda_i=\pm \sigma_i$$. Since $$\det(Q^T\Sigma)=\det(\Sigma)\le 0$$, at least one of the $$\lambda_i$$ must be non-positive. Thus, the smallest eigenvalue $$\lambda_1=-\sigma_1$$, and $$\lambda_i=\sigma_i$$ for $$i>1$$ (since we assumed that $$Q^T\Sigma$$ has at most one negative eigenvalue, and the absolute value of the negative eigenvalue is less than or equal to the next-smallest eigenvalue).

Thus, the eigenvalues of $$Q^T\Sigma$$ are the same as those of $$\Sigma$$.

Comment:

The strict inequality assumption $$\sigma_1 < \sigma_2$$ is necessary here:

If $$\sigma_1 = \sigma_2$$ one can take $$Q$$ to be diagonal, $$Q_{11}=Q_{22}=-1$$, and $$Q_{ii}=1$$ for $$i>1$$. Then, denoting $$\sigma=\sigma_1=\sigma_2$$, we get $$\Sigma=\text{diag}(-\sigma,\sigma,\sigma_3,\dots,\sigma_n), \, \, Q^T\Sigma=\text{diag}(\sigma,-\sigma,\sigma_3,\dots,\sigma_n)$$, and both satisfy the required condition on the eigenvalues.

I assume $$\sigma_1\neq 0$$, which means that $$\Sigma$$ is invertible. Let $$A=Q\Sigma^{-1}.$$ Then $$Q=A\Sigma$$ and $$Q^T=\Sigma A^T.$$

We want $$Q^T \Sigma$$ to be symmetric, which means $$Q^T\Sigma = \Sigma Q$$ or $$\Sigma A^T \Sigma = \Sigma A\Sigma.$$ If we multiply this with $$\Sigma^{-1}$$ from both sides, we get $$A=A^T,$$ so $$A$$ is symmetric.

$$Q$$ is orthogonal, which means $$Q^TQ=I$$ or $$\Sigma A^2 \Sigma = I$$ or $$A^2 = \Sigma^{-2}$$ So we are looking for a square root of $$\Sigma^{-2}$$ and the problem boils down to the question if $$\Sigma^{-1}$$ is the only valid choice.

We must consider the case that $$\Sigma$$ has eigenvalues with multiplicity of more than $$1.$$

Let $$\sigma_{r_i} = \sigma_{r_i+1} = \ldots = \sigma_{r_{i+1}-1}$$ for $$i=1,\ldots,m$$ and $$r_1=1,$$ $$r_2=2$$ and $$r_{m+1}=n+1.$$ Furthermore, $$\sigma_{r_i}<\sigma_{r_{i+1}}$$ for $$i=1,\ldots,m-1.$$ Then each square root of $$\Sigma^{-2}$$ can be written as follows $$A = \begin{pmatrix} \sigma_{r_1}^{-1} B_1 & & & & 0 \\ & \sigma_{r_2}^{-1} B_2 & & & \\ & & \sigma_{r_3}^{-1} B_3 & & \\ & & & \ddots & \\ 0 & & & & \sigma_{r_m}^{-1} B_m \end{pmatrix} \;\;,\;\; B_i^2 = I\;\;\mbox{for}\;\; i=1,\ldots,m$$ where $$B_i$$ are blocks of size $$(r_{i+1}-r_i)\times (r_{i+1}-r_i).$$ (The proof is given below)

Then $$Q = \begin{pmatrix} B_1 & & & & 0 \\ & B_2 & & & \\ & & B_3 & & \\ & & & \ddots & \\ 0 & & & & B_m \end{pmatrix}$$ The $$B_i$$ are symmetric. $$B_i^T$$ is the inverse of $$B_i$$ because of the orthogonality of $$Q$$, and $$B_i$$ is also the inverse of $$B_i$$, because of the property $$B_i^2=I.$$ Therefore $$B_i^T=B_i$$ and $$Q^T\Sigma = \begin{pmatrix} \sigma_{r_1}B_1 & & & & 0 \\ & \sigma_{r_2}B_2 & & & \\ & & \sigma_{r_3}B_3 & & \\ & & & \ddots & \\ 0 & & & & \sigma_{r_m}B_m \end{pmatrix}$$ We want $$Q^T\Sigma$$ to have the same eigenvalues as $$\Sigma,$$ which in turn means that $$\sigma_{r_i}B_i$$ has $$\sigma_{r_i}$$ as its only eigenvalue. A symmetric matrix with only one eigenvalue must be a scalar multiple of the identity matrix. Therefore, $$B_i = I$$ for $$i,\ldots,m,$$ which completes the proof.

Proof sketch for $$\sigma_1=0$$

If $$\sigma_1=0,$$ it can easily be shown that $$Q_{11}\in\{-1,1\}$$ and $$Q_{1j}=Q_{j1}=0$$ for $$j=2,\ldots,n.$$ This can be concluded from the symmetry of $$Q^T\Sigma$$ and from the orthogonality of $$Q.$$

This means that we can follow the argument from the first part of the proof, but consider only the subspace that is orthogonal to $$e_1.$$ Basically, this means that we ignore the first row and first column of all $$n\times n$$ matrices. In the end, we have to decide if $$Q_{11}=1$$ or $$Q_{11}=-1.$$ As $$Q\in \mathrm{SO}(n)$$ and $$B_i=I$$ for $$i=2,\ldots,m,$$ we can conclude $$Q_{11}=1.$$

Diagonalizable square roots of diagonal matrices

Let $$A$$ be diagonalizable and $$A^2$$ diagonal. Without loss of generality, the diagonal elements of $$A^2$$ are sorted in ascending order. Let $$0\leq\lambda_1 < \lambda_2 < \ldots < \lambda_m$$ such that the eigenvalues of $$A$$ form a (not necessarily strict) subset of $$\{\lambda_1,\;-\lambda_1,\;\lambda_2,\;-\lambda_2,\;\ldots,\;\lambda_m,\;-\lambda_m\}.$$ Let $$t_i^{+}$$ be the algebraic and geometric multiplicity of $$\lambda_i$$ and $$t_i^{-}$$ the algebraic and geometric multiplicity of $$-\lambda_i$$ within the matrix $$A$$ (we set $$t_1^{-}=0$$ if $$\lambda_1=0.$$) Let $$r_1=1$$ and $$r_{i+1} = r_i + t_i^{+}+ t_i^{-}.$$

If $$Av = \lambda v$$ and $$Aw = -\lambda w,$$ then $$A^2 (v+w) = A^2 v + A^2 w =\lambda^2 v + (-\lambda)^2 w = \lambda^2 (v+w).$$ This means that the eigenspace of $$A^2$$ with respect to the eigenvalue $$\lambda^2$$ is the direct sum of the eigenspaces of $$A$$ with respect to the eigenvalues $$\lambda$$ and $$-\lambda.$$

As $$A$$ is diagonalizable, the direct sum of the eigenspaces $$E_{A,\lambda_1},$$ $$E_{A,-\lambda_1}$$, $$E_{A,\lambda_2},$$ $$E_{A,-\lambda_2},\ldots$$, $$E_{A,\lambda_m},$$ $$E_{A,-\lambda_m}$$, forms the complete vector space $$\mathbb{R}^n.$$ This means that each of the eigenspaces of $$A^2$$ can be written as $$E_{A,\lambda_i} \oplus E_{A,-\lambda_i}.$$ In a manner of speaking, there is no room for other eigenspaces than those.

We know the eigenspaces of $$A^2,$$ because $$A^2$$ is diagonal. We have $$\begin{eqnarray*} E_{A^2,\lambda_1^2} & = & E_{A,\lambda_1} \oplus E_{A,-\lambda_1} = \mathrm{span}\{e_{r_1},\ldots,e_{r_2-1}\} \\ & \vdots & \\ E_{A^2,\lambda_m^2} & = & E_{A,\lambda_m} \oplus E_{A,-\lambda_m} = \mathrm{span}\{e_{r_m},\ldots,e_{r_{m+1}-1}\} \end{eqnarray*}$$ with the standard basis $$e_1,\ldots,e_n.$$ Now it is clear that $$A$$ can be diagonalized by means of a block matrix, because each $$E_{A,\lambda_i} \oplus E_{A,-\lambda_i}$$ is spanned by the related elements of the standard basis. $$A= \begin{pmatrix} S_1 & & 0 \\ & \ddots & \\ 0 & & S_m \end{pmatrix} \begin{pmatrix} \lambda_1 I_{t_1^{+}} & & & & 0 \\ & -\lambda_1 I_{t_1^{-}} & & & \\ & & \ddots & & \\ & & & \lambda_m I_{t_m^{+}} & \\ 0 & & & & -\lambda_m I_{t_m^{-}} \end{pmatrix} \begin{pmatrix} S_1 & & 0 \\ & \ddots & \\ 0 & & S_m \end{pmatrix} ^{-1}$$ From this, by simply processing the matrix multiplication, we can conclude that $$A$$ itself is also a block matrix of the same sort, i.e. $$A = \begin{pmatrix} A_1 & & 0 \\ & \ddots & \\ 0 & & A_m \end{pmatrix}$$ with $$A_i = S_i\,\begin{pmatrix} \lambda_i I_{t_i^{+}} & \\ & -\lambda_i I_{t_i^{-}} \\ \end{pmatrix} \, S_i^{-1}$$ Now we only have to show that $$A_i = \lambda_i B_i$$ with $$B_i^2=I.$$

Let $$T_i=S_i^{-1}$$.

Let $$S_i^{+}$$ be the $$(t_i^{+}+t_i^{-})\times t_i^{+}$$ matrix that is formed by the first $$t_i^{+}$$ columns of $$S_i$$ and $$S_i^{-}$$ the $$(t_i^{+}+t_i^{-})\times t_i^{-}$$ matrix that is formed by the last $$t_i^{-}$$ columns of $$S_i.$$ Let $$T_i^{+}$$ be the $$t_i^{+}\times (t_i^{+}+t_i^{-})$$ matrix that is formed by the first $$t_i^{+}$$ rows of $$T_i$$ and $$T_i^{-}$$ the $$t_i^{-}\times (t_i^{+}+t_i^{-})$$ matrix that is formed by the last $$t_i^{-}$$ rows of $$T_i.$$

Then $$T_i^{+}S_i^{+}=I,\;\;T_i^{-}S_i^{-}=I,\;\;T_i^{+}S_i^{-}=0,\;\;T_i^{-}S_i^{+}=0$$. $$A_i = S_i^{+}\lambda_i T_i^{+} + S_i^{-}(-\lambda_i) T_i^{-} = \lambda_i \left( S_i^{+}T_i^{+} - S_i^{-}T_i^{-}\right)$$ Let $$B_i = S_i^{+}T_i^{+} - S_i^{-}T_i^{-}.$$ Then $$\begin{eqnarray*} B_i^2 & = & \left( S_i^{+}T_i^{+} - S_i^{-}T_i^{-}\right)\left( S_i^{+}T_i^{+} - S_i^{-}T_i^{-}\right) \\ & =& S_i^{+}T_i^{+}S_i^{+}T_i^{+}-S_i^{+}T_i^{+}S_i^{-}T_i^{-}-S_i^{-}T_i^{-}S_i^{+}T_i^{+}+S_i^{-}T_i^{-}S_i^{-}T_i^{-} \\ & =& S_i^{+}\cdot I\cdot T_i^{+}-S_i^{+}\cdot 0 \cdot T_i^{-}-S_i^{-}\cdot 0 \cdot T_i^{+}+S_i^{-}\cdot I\cdot T_i^{-} \\ & =& S_i^{+}T_i^{+}+S_i^{-}T_i^{-} \\ & =& \begin{pmatrix} & & \\ S_i^{+} & & S_i^{-} \\ & & \end{pmatrix} \begin{pmatrix} & T_i^{+} & \\ & & \\ & T_i^{-} & \end{pmatrix} =S_iT_i = I \end{eqnarray*}$$

• Thanks, this is very nice. I have two questions: (1) Can you elaborate on the known fact about square roots of diagonal matrices? How do we know that every square root is of the form you have prescribed? (we are actually looking only at symmetric square roots of $\Sigma^{-2}$, since you already showed that $A$ is symmetric. Thus we can use orthogonal diagonalization, but I don't see how it gets us to the form you specified). – Asaf Shachar Dec 31 '19 at 14:12
• (2) Just to make sure I understand: The part where you use the assumption that $\sigma_1 < \sigma_2$ is in the claim that " $Q^T\Sigma$ has the same eigenvalues as $\Sigma,$ implies that $\sigma_{r_i}B_i$ has $\sigma_{r_i}$ as its only eigenvalue.", right? Otherwise, if $\sigma_2=-\sigma_1$, we can take $Q_{11}=Q_{22}=-1$ as I commented in the body of the question. – Asaf Shachar Dec 31 '19 at 14:12
• (1) I added the required proof. I had forgotten to mention that it holds for diagonalizable square roots, only, which is good enough in this case. (2) Yes, this is correct. – Reinhard Meier Jan 3 '20 at 12:14