Taylor expansion of a function of a matrix

Question

Assume that ${\mathbf H}$ is a $N \times M$ matrix. The following parameter is called orthogonality deficiency and describes how much orthogonal the columns of ${\mathbf H}$ are. $$ od({\mathbf H}) = 1 - \frac{\det({{\mathbf H}^H{\mathbf H})}}{\Pi_{n=1}^M\|{{\mathbf h}_n}\|^2}$$ where ${\mathbf h}_n$ is the $n$th column of matrix ${\mathbf H}$. It can be seen that $0\leq od({\mathbf H})\leq 1$. If ${\mathbf H}$ is singular then $od({\mathbf H})=1$ and when $od({\mathbf H})=0$ the colums of mathrix ${\mathbf H}$ are orthogonal. This is, indeed a very useful criterion for matrix orthogonality and has many applications in communications engineering and signal processing.

My question:

When the matrix ${\mathbf H}$ has the form ${\mathbf H}={\mathbf A}+e{\mathbf B}$, where $e$ is a scalar, how can we approximate $od({\mathbf H})$ when $e \ll 1$ and what is the approximation of $od({\mathbf H})$ when $e \ll 1$ (Probably using Taylor expansion)?

Answer 1

If $e$ is a small number, then $od(H)\approx od(A)+e(od)'_A(B)$. $od(A)=1-\dfrac{u(A)}{v(A)}$ where $u(A)=\det(A^*A),v(A)=\Pi_i||Ae_i||^2$ and $(e_i)_i$ is the canonical basis. $(od)'_A=-\dfrac{1}{v(A)}u'_A+\dfrac{u(A)}{v^2(A)}v'_A$. $u'_A(K)=trace((A^*K+K^*A)adjoint(A^*A))$. $v'_A(K)=\sum_i((e_i^*K^*Ae_i+e_i^*A^*Ke_i)\Pi_{j\not= i}||Ae_j||^2)$.

Edit: 1. The proof is based on this fact: if $\phi:A\rightarrow \det(A)$, then $\phi'_A:K\rightarrow trace(K.adjoint(A))$. 2. Simplifications: $u(A)=|\det(A)|^2$ and $e_i^*K^*Ae_i+e_i^*A^*Ke_i=2Real((K^*A)_{i,i})$. 3. In the sake of simplicity, assume that $K$ is real. Let $(E_{i,j})$ be the canonical basis of $\mathcal{M}_n(\mathbb{R})$. Then the matrix of $(od)'_A$ is in the form $U=[u_1,\cdots,u_{n^2}]$. If you want a large variation of $od(A)$, then choose $B=U^T$ (in the orthogonal of $\ker(od'_A)$). Then $(od)'_A(B)=||U||^2$. For instance, if $A=\begin{pmatrix}8&-5-6I&-3-I\\-5+4I&2+3I&7+7I\\6-I&-4+6I&8+2I\end{pmatrix}$, then $U\approx [-0.0029,-0.0154,-0.0120,-0.0010,-0.0064,-0.0119,-0.0110,-0.0111,0.0124]$. $od(A)\approx 0.9489$ and $od(A-100.U^T)\approx 0.8147$.