# Tagged Questions

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### To prove that $A$ has a one-dimensional eigenspace , where $A \in SO(3)$ , $A \ne I$

Let $A\ne I$ be a $3\times3$ real orthogonal matrix with determinant $1$ , then how to prove that $A$ has a one-dimensional eigenspace ?
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### proving bounds by bounding eigenvalues?

I have an expression that takes the form:$\mu_n= \displaystyle (I_d+\sum_{i=1}^n f_i f_i^T)^{-1}(\displaystyle\sum_{i=1}^n f_i)$ where $f_i$'s are d dimensional vectors with bounded $L^2$ norm. $I_d$ ...
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### Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix}$$ I have found the eigenvalues to be 1 ...
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### How prove this matrix $B^{-1}-A^{-1}$ is positive-semidefinite matrix,if $A-B$ is positive matrix

Question: Let $A,B$ be positive $n\times n$ matrices, and assume that $A-B$ is also a positive definite matrix. Show that $$B^{-1}-A^{-1}$$ is a positive definite matrix too. My idea: ...
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