If you take a tangent of a matrix, how can you visualise what is happening geometrically? E.g. for
$$ \tan \theta =3 $$
You could envision a right triangle with an angle $\theta$ and a opposite side that is $3$ times bigger than the adjacent. Now imagine taking the tangent of a matrix with the use of a power series:
$$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$$
So for matrixes, e.g.
$$ \tan \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$$
:
$$\tan A = A + \dfrac 1 3 A^3 + \dfrac 2 {15} A^5 + \dfrac {17} {315} A^7 + \dfrac {62} {2835} A^9 + \cdots$$
Is there any way to connect this to triangles etcetera?
Best Answer
tl; dr: Modulo similarity, this amounts to little more than considering matrices having values of tangent on the diagonal. This construction therefore has a geometric interpretation to the same extent that one exists for $1 \times 1$ real matrices.
$\DeclareMathOperator{\diag}{diag}$Throughout, let $(\lambda_{j})_{j=1}^{n}$ be an ordered set of complex numbers (e.g., real numbers). If $$ \Lambda = \diag[\lambda_{j}] = \diag[\lambda_{1}\ \lambda_{2}\ \cdots\ \lambda_{n}], $$ then for every non-negative integer $k$ we have $$ \Lambda^{k} = \diag[\lambda_{j}^{k}] = \diag[\lambda_{1}^{k}\ \lambda_{2}^{k}\ \cdots\ \lambda_{n}^{k}]. $$ To each formal power series $f(x) = \sum_{k} c_{k}x^{k}$, we can associate the formal matrix series \begin{align*} f(\Lambda) &= \sum_{k=0}^{\infty} c_{k} \Lambda^{k} \\ &= \sum_{k=0}^{\infty} c_{k} \diag[\lambda_{j}^{k}] \\ &= \diag\biggl[\sum_{k=0}^{\infty} c_{k} \lambda_{j}^{k}\biggr] \\ &= \diag[f(\lambda_{1})\ f(\lambda_{2})\ \cdots\ f(\lambda_{n})]. \end{align*} If the series for $f$ converges with positive radius $r$, then for every diagonal matrix $\Lambda = [\lambda_{j}]$ whose entries satisfy $|\lambda_{j}| < r$ for $1 \leq j \leq n$, the preceding formula may be interpreted "properly," as evaluating the analytic function $f$ on the matrix $\Lambda$ and obtaining the diagonal matrix formed from evaluating $f$ on the eigenvalues (diagonal entries) of $\Lambda$.
Further, if $P$ is invertible, then the identity $(P\Lambda P^{-1})^{k} = P\Lambda^{k}P^{-1}$ implies $$ f(P\Lambda P^{-1}) = Pf(\Lambda)P^{-1}. $$ Consequently, if $A$ is diagonalizable and $\Lambda = P^{-1}AP$ is diagonal (so $A = P\Lambda P^{-1}$), the preceding discussion allows us to define and calculate $f(A)$, properly if every eigenvalue of $A$ has absolute value smaller than $r$, and formally otherwise.
Entirely similar ideas allow us to evaluate $f$ on Jordan block matrices. Since every complex matrix is similar to a Jordan block matrix, this discussion defines $f(A)$ as a formal series for all $A$, and as a proper function for matrices whose eigenvalues have sufficiently small magnitude.