Prove that the following application between $M_{n×n}(\Bbb R)$ and $M_{n^2×n^2}(\Bbb R)$ is an isometry such that $\det φ(X)≠ 0$ whether $\det X≠ 0$

Let be $\varphi:M_{n\times n}(\Bbb R)\rightarrow M_{n^2\times n^2}(\Bbb R)$ the application defined through the equation $$ \varphi(X):=\sum_{i,j,h=1}^{n}x^{i,j}\hat e_{h+(i-1)n}\otimes\hat e_{j+(h-1)n} $$ for any $X\in M_{n\times n}(\Bbb R)$. So I ask me if $\varphi(X)$ is invertible when $X$ is and moreover observing that $\varphi$ is a linear application I ask also to me if $\varphi $ is an isometry. So first of all I suppose $n=3$ so that $$ \varphi(X)=\begin{pmatrix}x_{1,1} && x_{1,2} && x_{1,3} && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && x_{1,1} && x_{1,2} && x_{1,3} && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && x_{1,1} && x_{1,2} && x_{1,3} \\x_{2,1} && x_{2,2} && x_{2,3} && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && x_{2,1} && x_{2,2} && x_{2,3} && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && x_{2,1} && x_{2,2} && x_{2,3} \\x_{3,1} && x_{3,2} && x_{3,3} && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && x_{3,1} && x_{3,2} && x_{3,3} && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && x_{3,1} && x_{3,2} && x_{3,3} \end{pmatrix} $$ for any $X\in M_{3\times 3}(\Bbb R)$ and thus I observed that $$ \varphi(X)\cdot\big(\varphi(Y^T)\big)^T=\begin{pmatrix}x_{1,1} && x_{1,2} && x_{1,3} && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && x_{1,1} && x_{1,2} && x_{1,3} && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && x_{1,1} && x_{1,2} && x_{1,3} \\x_{2,1} && x_{2,2} && x_{2,3} && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && x_{2,1} && x_{2,2} && x_{2,3} && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && x_{2,1} && x_{2,2} && x_{2,3} \\x_{3,1} && x_{3,2} && x_{3,3} && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && x_{3,1} && x_{3,2} && x_{3,3} && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && x_{3,1} && x_{3,2} && x_{3,3} \end{pmatrix}\cdot\begin{pmatrix}y_{1,1} && 0 && 0 && y_{1,2} && 0 && 0 && y_{1,3} && 0 && 0 \\y_{2,1} && 0 && 0 && y_{2,2} && 0 && 0 && y_{2,3} && 0 && 0 \\y_{3,1} && 0 && 0 && y_{3,2} && 0 && 0 && y_{3,3} && 0 && 0 \\0 && y_{1,1} && 0 && 0 && y_{1,2} && 0 && 0 && y_{1,3} && 0 \\0 && y_{1,2} && 0 && 0 && y_{2,2} && 0 && 0 && y_{2,3} && 0 \\0 && y_{3,1} && 0 && 0 && y_{3,2} && 0 && 0 && y_{3,3} && 0 \\0 && 0 && y_{1,1} && 0 && 0 && y_{1,2} && 0 && 0 && y_{1,3} \\0 && 0 && y_{2,1} && 0 && 0 && y_{2,2} && 0 && 0 && y_{2,3} \\0 && 0 && y_{3,1} && 0 && 0 && y_{3,2} && 0 && 0 && y_{3,3} \end{pmatrix}=\dots=\varphi(X\cdot Y^T) $$ from which the statement follows observing that $$ \text{rank}\,\varphi(I)=\text{rank}\,\begin{pmatrix}1 && 0 && 0 && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && 1 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && 1 && 0 && 0 \\0 && 1 && 0 && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 1 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && 0 && 1 && 0 \\0 && 0 && 1 && 0 && 0 && 0 && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 1 && 0 && 0 && 0 \\0 && 0 && 0 && 0 && 0 && 0 && 0 && 0 && 1 \end{pmatrix}=9 $$ because this implies that $$ \det\varphi(X)\cdot\det\varphi(X^{-T})=\det\big(\varphi(X)\cdot\varphi(X^{-T})\big)=\det\big(\varphi(X\cdot X^{-1})\big)=\det\big(\varphi(I)\big)\neq 0 $$ from which by the zero-product property I conclude that the quantity $\det\big(\varphi(X)\big)$ is not zero. Moreover observing that $$ \text{tr}\big(\varphi(X)\big)=\text{tr}(X) $$ for any $X\in M_{3\times 3}(\Bbb R)$ I conclude that $$ \big\langle\varphi(X),\varphi(Y)\big\rangle=\text{tr}\big(\varphi(X)\cdot\varphi(Y)^T\big)=\\\text{tr}\Big(\varphi(X)\cdot\varphi\big((Y^T)^T\big)^T\Big)=\text{tr}\big(\varphi(X\cdot Y^T)\big)=\text{tr}\big(X\cdot Y^T\big)=\langle X,Y\rangle $$ so that $\varphi$ is an isometry. So the statement is srely true for $n=3$, provided that the argumetations I gave are correct. Anyway I think that the statement, at least the first part, can be proved using the following conjecture which I did not able to prove

CONJECTURE

Any isometry $\phi$ between $M_{n\times n}(\Bbb R)$ and $M_{n^2\times n^2}(\Bbb R)$ is such that $$ \det\big(\phi(X)\big)\neq 0 $$ provided that $X$ is not singular.

Anyway I think that it is better to pose a dedicated question to this.

So I ask if the argumetation I gave are correct and if can be generalised to an arbitrary $n\in\Bbb N$. So could anyone help me, please?

Solution 1:

At the risk of being a bit confusing, I will use $\otimes$ to denote the Kronecker product. We have $$ \varphi(X) = \pmatrix{X \otimes e_1 & X \otimes e_2 & \cdots & X \otimes e_n}. $$ From there, we find that $$ \langle \varphi(X), \varphi(Y) \rangle = \sum_{i=1}^n \langle X \otimes e_i, Y \otimes e_i\rangle = \sum_{i=1}^n \langle X,Y \rangle = n \langle X, Y \rangle. $$ So, it is not true that $\varphi$ is an isometry. However, $\psi(X) := \varphi(X)/n$ is an isometry.

Regarding the comment: note that $$ \varphi(X) \varphi(Y^T)^T = \\ \pmatrix{X \otimes e_1 & \cdots & X \otimes e_n} \pmatrix{Y \otimes e_1^T \\ \vdots \\ Y \otimes e_n^T} = \\ (X \otimes e_1)(Y \otimes e_1^T) + \cdots + (X \otimes e_n)(Y \otimes e_n^T) =\\ (XY) \otimes (e_1e_1^T) + \cdots + (XY) \otimes (e_ne_n^T) = \\ (XY) \otimes (e_1e_1^T + \cdots + e_ne_n^T) = \\ (XY) \otimes I. $$ Note that in general, if $A$ is $m \times n$ with entries $a_{ij}$, then $$ A \otimes I = \pmatrix{ a_{11} I & \cdots & a_{1n} I \\ \vdots & \ddots & \vdots \\ a_{m1} I & \cdots & a_{mn} I}. $$