2
$\begingroup$

How to find an example of matrix $A$ that satisfies $A^{-1} = \frac{1}{n} A$, where $A \in n \times n$? For example if $A= \left( \begin{array}{ccc} 1 & 1 & 1\\ 1 & i & i^2\\ 1 & i^2 & i^4 \\ \end{array} \right)$, then $A^{-1}=\frac{1}{3} \overline{A} $

  • 0
    I meant that $A = [a_{ij}]_{n \times n}$2012-11-27

3 Answers 3

1

Hadamard matrices all have this property.

3

The easiest way is to let $A$ be a multiple of the identity. These are easy to invert.

3

Your matrix satisfies the polynomial $A^2 - nI = 0$. Therefore the minimal polynomial of your matrix is either $A \pm \sqrt{n}I = 0$ where the only matrices which satisfy the criteria are $A=\pm\sqrt{n}I$ or the minimal polynomial splits as $(A+\sqrt{n}I)(A-\sqrt{n}I) = 0$. In this case, you can take any matrix similar to a diagonal matrix with $\pm\sqrt{n}$ on the diagonals.

  • 0
    Ok, please do not re-define existing notation without at least telling us what it means. There is no one who could've possibly guessed what you wanted $A^2$ to be. If you want your $"A^2"$ to equal $A$, then each entry needs to satisfy $a^2 = a$. Each entry therefore needs to be $0$ or $1$.2012-11-27