As we know we can construct unitary matrix as $H=H\_1H\_2\dots,H\_k$, by stacking householder matrices $H\_i\in \mathbb{R}^{d\times d}$. The number of householder matrices we use, i.e., $k$, determines the expressivity of construction.


Conversely, for a given unitary matrix $H$, how can I know how many householder matrices I need at least for building this $H$? I mean how can I know the minimal $k$.



How many householder matrices do I need for constructing a given unitary matrix?
A Householder decomposition of a $d \times d$ unitary $H$ can be achieved with *at most* $d$ Householder matrices (see [Theorem 1](https://core.ac.uk/download/pdf/82680205.pdf)) and a tighter upper bound is $d-m$ Householder matrices where $m = \dim( \operatorname{ker}(H-I\_d) )$ (see [Theorem 2](https://core.ac.uk/download/pdf/82680205.pdf)).


*Uhlig, Frank*, [**Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries**](http://dx.doi.org/10.1016/S0024-3795(01)00296-8), Linear Algebra Appl. 332-334, No. 1-3, 459-467 (2001). [ZBL0982.65049](https://zbmath.org/?q=an:0982.65049).


