When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Question

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$).

I was wondering when the inner product $ \langle gI, t \rangle$ is equal to $ \langle I, g^{-1} t\rangle $ i.e.:

$$ \langle gI, t \rangle = \langle I, g^{-1} t\rangle$$

What kind of space does it have to be? What kind of group (action) does $g$ has to have? What type of conditions does the inner product have to have? Any conditions for that to hold...or maybe that equation can never hold?

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These are some of my thoughts:

To get familiar with the problem I considered the dot product (for the inner product) and examined both sides of the equation:

$$ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $$

$$ \sum^{d}_{i=1} (gI)_i t_i = \sum^{d}_{i=1} I_i (g^{-1} t)_i $$

however, doing that didn't reveal that much to me. My intuition says that maybe cyclic groups might be import, though not sure if they are.

Some other idea I have, maybe leaving it in terms of dot products rather than expanding the summation leads to:

$$(gI)^T t = I^T g^{-1}t$$

$$I^T g^T t = I^T g^{-1}t$$

so $g^{T} = g^{-1}$? Seems strange to me.

Answer 1

Suppose that $V$ is an inner product space. Let $T:V \to V$ be a linear transformation. What is known as the adjoint of $T$, denoted by $T^*$, has the defining characteristic $$ \langle T(\textbf{v}),\textbf{w}\rangle=\langle\textbf{v},T^*(\textbf{w})\rangle , \mbox{ for all }\textbf{v},\textbf{w}\in V.$$ Furthermore, $T$ is referred to as unitary precisely when $TT^*=T^*T=id_V$. That is, when $T^* = T^{-1}$. More information on adjoints can be found here.

So, it is clear that unitary transformations in $\mathbb{R}^d$ will satisfy your relation. It should be noted that when we restrict $V$ to be $\mathbb{R}^d$, we tend to call the transformations orthogonal instead of unitary.

Furthermore, when we restrict $V$ to be finite-dimensional, the notion of a unitary linear transformation is equivalent to an isometric linear transformation. More information on (linear) isometries can be found here.