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)? 



Taylor expansion of a function of a matrix
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 $od(A-100.U^T)\approx 0.8147$.


