I have proofed this orthogonal property. Please correct it or show your version of the proof if I am wrong: $A^{-1} = A^t$ $A^{-1} \times A = A^t \times A$ $I = I$ I appreciate your answer

I want to generate $500$ random integer matrices with integer eigenvalues. Thanks to this post, I know how to generate a random matrix with whole eigenvalues: Generate a diagonal matrix $D$ with the desired (integer) eigenvalues. Generate an invertible matrix $A$ of the same size as $D$. Record the matrix $A^{-1} D A$. However, the […]

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm not greater than one? It is easy to show that a convex combination of orthogonal matrices has norm (I mean the norm as operators) not larger than $1$. The […]

This question already has an answer here: Compactness of the set of $n \times n$ orthogonal matrices 1 answer

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices. For example, if I have a linear transformation, say on the plane, my intuition is […]

Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.

Let $A \in M_n(\Bbb R)$. How can I prove, that 1) if $ \forall {b \in \Bbb R^n}, b^{t}Ab>0$, then all eigenvalues $>0$. 2) if $A$ is orthogonal, then all eigenvalues are equal to $-1$ or $1$

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