Let $A \in \mathbb{C}^{n \times n}$ and for a given $x \in \mathbb{C}^n$, the square gain of $A$ is $\lVert A x \rVert^2 / \lVert x \rVert^2$ where the norm is the usual Euclidean norm. I want to calculate the average gain over all possible $x \in \mathbb{C}^n$. Obviously, the gain is independent from the "length" of $x$, so without losing generality we can assume $\lVert x \rVert = 1$.

Now we need a method to parameterize all $\lVert x \rVert = 1$. The first thing that comes to my mind is using the singular value decomposition of $A$. Let $A = U \Sigma V^*$ be the SVD of $A$, so

$$x = \sum_{i=1}^n \alpha_i v_i ~~~~\text{such that}~~~~ \sum_{i=1}^n \alpha_i^2 = 1 $$

characterize all possible vectors, where $U = [u_1 ~~ \dots ~~ u_n], \Sigma = \operatorname{diag}\{\sigma_i\}$ and $V = [v_1 ~~ \dots ~~ v_n]$.

Let's first consider $n=2$. We can select $\alpha_1 := \cos \theta$ and $\alpha_2 := \sin \theta$. So, the average gain can be calculated as

$$\begin{align*} \frac{1}{2 \pi} \int_0^{2\pi} \lVert A v_1 \cos \theta + A v_2 \sin \theta \rVert^2 d\theta &= \frac{1}{2 \pi} \int_0^{2\pi} \lVert \sigma_1 u_1 \cos \theta + \sigma_2 u_2 \sin \theta \rVert^2 d\theta \\ &= \frac{1}{2 \pi} \int_0^{2\pi} \left( \sigma_1^2 \cos^2 \theta + \sigma_2^2 \sin^2 \theta \right) d\theta \\ &= \frac{\sigma_1^2 + \sigma_2^2}{2} \end{align*}$$

This result makes intuitive sense since the average square gain is the average of minimum and maximum square gains. So, we can hypotesize that the average square gain is $\lVert A \rVert_F^2 / n$ where $\lVert \cdot \rVert_F$ is the Frobenius norm. Now we can test this idea for $n=3$ using spherical coordinates. So let $\alpha_1 := \cos \theta, \alpha_2 := \sin \theta \cos \phi, \alpha_3 := \sin \theta \sin \phi$. Using the same steps we get

$$\begin{align*} &\frac{1}{2 \pi^2} \int_0^{\pi} \int_0^{2\pi} \lVert A v_1 \cos \theta + A v_2 \sin \theta \cos \phi + A v_3 \sin \theta \sin \phi \rVert^2 d\phi d\theta \\ &~~~~~~~~= \frac{1}{2 \pi^2} \int_0^{\pi} \int_0^{2\pi} \left( \sigma_1^2 \cos^2 \theta + \sigma_2^2 \sin^2 \theta \cos^2 \phi + \sigma_3^2 \sin^2 \theta \sin^2 \phi \right) d\phi d\theta \\ &~~~~~~~~= \frac{2\sigma_1^2 + \sigma_2^2 + \sigma_3^2}{4} \end{align*}$$

which is not what we expected. Coming back to $n=2$ case, we could have also selected $\alpha_1 = t$ and $\alpha_2 = \sqrt{1-t^2}$ where $t \in [0,1]$. In this case, we get

$$ \int_0^1 \left(\sigma_2^2 + (\sigma_1^2 - \sigma_2^2) t^2 \right) dt = \frac{\sigma_1^2 + 2 \sigma_2^2}{3} $$

which is different from our first attempt.

So, why are the results are different? Is there a "correct" way of selecting the parameters to get a "nice" formula like $\lVert A \rVert_F^2 / n$?

  • $\begingroup$ The first parametrization induces a uniform measure (Haar measure) on the unit circle, the second one doesn't. $\endgroup$ Jan 2 at 20:39
  • $\begingroup$ Possibly related: this and this and this $\endgroup$ Jan 2 at 21:54
  • 1
    $\begingroup$ Rewrite the problem as $$ \frac{\|Ax\|^2}{\|x\|^2} \;=\; A^TA:\left\langle \frac{xx^T}{x^Tx}\right\rangle \;=\; A^TA:\oint nn^T d\Omega $$ Then utilize the results of this answer to confirm your initial result $$ \frac{\|Ax\|^2}{\|x\|^2} \;=\; A^TA:\left(\frac In\right) \;=\; \frac{A:A}{n} \;=\; \frac{\big\|A\big\|_F^2}{n} $$ The caveat is that the linked post is for ${\mathbb R}^n \;$ $\endgroup$
    – greg
    Jan 4 at 9:42

1 Answer 1


Equip $S^{n - 1} = \{x \in \mathbb{R}^n : |x| = 1\}$ with it's usual surface measure, normalized to be a probability measure. Then you seek $\int_{S^{n - 1}}|Ax|^2\,dS(x)$.

As for selecting parameters, any parameterization of $S^{n - 1}$ will work. You have to apply the formula for the surface integral though:

If $\phi : O \to U \subset {S}^{n - 1}$ is a $C^1$ parameterization and $f : S^{n - 1} \to \mathbb{R}$ vanishes off $U$, then $$\int_{U}f(x)\,dS(x) = \frac{1}{\text{Area}(S^{n - 1})}\int_{O}f(\phi(x))\sqrt{\det D\phi(x)^T D\phi(x)}\,dx.$$ I think you forgot the $\sqrt{ \det D\phi(x)^T D\phi(x)}$. Luckily, this problem is simple enough that we don't need to use coordinates.

Write the SVD of $A$ as $A = UDV^T$. Then $|Ax|^2 = |DV^Tx|^2$. So by invariance of the measure under orthogonal transformations, $$\int_{S^{n - 1}}|Ax|^2\,dS(x) = \int_{S^{n - 1}}|DV^Tx|^2\,dS(x) = \int_{S^{n - 1}}|Dx|^2\,dS(x).$$ Now note $|Dx|^2 = (D^2x, x) = \sigma_1^2|x_1|^2 + \dots + \sigma_n^2|x_n|^2$. Hence $$\int_{S^{n - 1}}|Dx|^2\,dS(x) = \sum_{i = 1}^{n}\sigma_i^2\int_{S^{n - 1}}|x_i|^2\,dS(x).$$ Note that $a_i = \int_{S^{n - 1}}|x_i|^2\,dS(x)$ is independent of $i$ by invariance of the measure under orthogonal transformations. Moreover, $$a_1 + \dots + a_n = \int_{S^{n - 1}}|x|^2\,dS(x) = 1.$$ Hence $a_i = \frac{1}{n}$. In conclusion, $$\int_{S^{n - 1}}|Ax|^2\,dS(x) = \frac{\sigma_1^2 + \dots + \sigma_n^2}{n}.$$

Edit: Here is why $a_i = \int_{S^{n - 1}}|x_i|^2 dS(x)$ is independent of $i$. Fix $i$. Note that \begin{align} \int_{S^{n - 1}}|x_i|^2\,dS(x) &= \int_{S^{n - 1}}|(x, e_i)|^2\,dS(x). \end{align} Now pick an orthogonal transformation $R$ such that $Re_i = e_1$. Since $(x, e_i) = (Rx, Re_i) = (Rx, e_1)$, it follows that $$\int_{S^{n - 1}}|x_i|^2\,dS(x) = \int_{S^{n - 1}}|(Rx, e_1)|^2\,dS(x) = \int_{S^{n - 1}}|(x, e_1)|^2\,dS(x),$$ Where the last equality follows from invariance of the measure under orthogonal transformations. So $a_i = a_1$.

The definition of the "surface measure" I am using here is the one that works for any $C^1$ $m$-dimensional surface $M \subset \mathbb{R}^k$. The measure $\mu$ is a Borel measure on $M$ defined in such a way that if $f : M \to [0, \infty]$ is a measurable function vanishing off the image $U \subset M$ of a coordinate parameterization $\phi : O \to U$, $O$ open in $\mathbb{R}^m$, then $$\int_{M}f\,d\mu = \int_{U}f\,d\mu = \int_{O}f(\phi(x))\sqrt{g(x)}\,dx, \hspace{20pt} g(x) = \det D\phi(x)^T D\phi(x).$$ Here I used it for $M = S^{n - 1}$ (I also normalized it to $1$). From this you can prove the assertion that the measure on $S^{n - 1}$ is invariant under orthogonal transformations, that is, if $R^TR = I$, and $f : S^{n - 1} \to [0, \infty]$ is measurable, then $$\int_{S^{n - 1}}f(Rx)\,d\mu(x) = \int_{S^{n - 1}}f(x)\,d\mu(x).$$ A way to prove this is to first prove it for $f$ vanishing off of a coordinate patch, in which case it is just a computation. Then the fact that $M$ is always a countable union of such patches implies the general case.

The above measure is also defined on a general Riemannian manifold $M$, in which case $D\phi(x)^T D\phi(x)$ is replaced with the coordinate matrix $G(x)$ of the metric tensor of $M$ (the above definition is the special case when the metric tensor is the dot product).

  • $\begingroup$ After some reading about surface measure and surface integral I understand the answer and it is clear, so thanks. As I understand it $dS$ has to be "uniform" under the selected parameterization (Haar measure?). But as you showed that we don't need to specify the parameterization explicitly for this problem as long as we assume it is uniform. I also understand intuitively that all $a_i$ must be equal. But I'm having trouble showing it rigorously, could you elaborate on this point? $\endgroup$
    – obareey
    Jan 3 at 15:05
  • $\begingroup$ @obareey You can use any parameterization for integrating, as long as you apply the coordinate formula for integration. I added elaboration on the surface measure. It is defined on any Riemannian manifold. The word "uniform" seems to be synonymous with the surface measure. Haar measure is a different concept than surface measure, though it is a relevant in relation to the surface measure on the sphere. $\endgroup$
    – Mason
    Jan 4 at 21:37

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