Requirement That a Vector be Related to Itself Through Identity

Question

If I have two vectors for which the relation can be written $$ \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} = \begin{bmatrix}A\end{bmatrix} \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} $$ Is [A] necessarily an identity matrix?

What if I have a condition that $\vec{I}_1$ = 0.5*($\vec{I}_2$ + $\vec{I}_3$), can this not be phrased as: $$ \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} = \begin{bmatrix} 0&0.5&0.5\\ 2&0&-1\\ 2&-1&0\\ \end{bmatrix} \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} $$

Does this stipulate that somehow

$$ \begin{bmatrix} 0&0.5&0.5\\ 2&0&-1\\ 2&-1&0\\ \end{bmatrix} = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix} $$

I am of course aware that the left hand matrix can be rearranged to get the right hand matrix. Clarification on my bad assumptions would be appreciated - does this mean a set of vectors does not necessarily need to be equal to itself?

Thanks

Answer 1

Not quite. The fact that if $v = \left[\begin{array}{c}x_1\\x_2\\x_3 \end{array}\right]$ and $Av=v$, then $v$ is an eigenvector for $A$ with corresponding eigenvalue 1. For instance, if $v = \left[\begin{array}{c}2\\0\\0 \end{array}\right]$ and $A = \left[\begin{array}{ccc}1 & 0 & 0 \\0 &0 & 0 \\0 & 0 & 0 \end{array}\right]$, then $Av = v$. However, if $Av = v$ for all $v \in \mathbb{R}^3$, then $A = I_{3\times 3}$ the $3\times 3$ identity matrix.

Answer 2

If this is true for all $I_1$, $I_2$, $I_3$, then yes, $A$ must be the identity matrix.

But, as you have shown, if it is true for a specific $I_1$, $I_2$, $I_3$, there are many things $A$ could be, not just the identity. The two matrices that you have aren't equal, but they act on $I_1$, $I_2$, $I_3$ the same way. If you want an analogy, the functions $f(x) = x$ and $f(x) = x^2$ act on $0$ (and $1$) the same way, but they certainly are not the same function.

As for "rearranging" (I assume you mean row-reduction): just because two matrices can be turned into each other, it doesn't mean that they are equal. Unfortunately, row-reduction is often taught as "rewriting" the matrix, as if it were just a different representation. But in reality, you're writing an entirely new matrix.