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Let $A$ be a nondefective $n \times n$ matrix with diagonalizing matrix $X .$ Show that the matrix $Y=\left(X^{-1}\right)^{T}$ diagonalizes $A^{T}$...

Question

Let $A$ be a nondefective $n \times n$ matrix with diagonalizing matrix $X .$ Show that the matrix $Y=\left(X^{-1}\right)^{T}$ diagonalizes $A^{T}$

Let $A$ be a nondefective $n \times n$ matrix with diagonalizing matrix $X .$ Show that the matrix $Y=\left(X^{-1}\right)^{T}$ diagonalizes $A^{T}$



Answers

Let $\mathbf{A}$ be an $m \times n$ matrix. Show that $\mathbf{A}^{T} \mathbf{A}$ is a symmetric
$n \times n$ matrix and $\mathbf{A} \mathbf{A}^{T}$ is a symmetric $m \times m$ matrix (see
Problem 41$).$


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