A matrix ๐ด in ๐๐(โ) is called skew-symmetric if ๐ดแต = โ๐ด.
Show that the set of all skew-symmetric matrices in ๐๐(โ) is a subspace of ๐๐(โ) and determine its dimension (in term of n ).
Let ๐ด โ ๐๐(โ). Prove that vแต๐ด๐ฃ = 0 for ๐ฃ โ โ๐ if and only if ๐ด is skew-symmetric.
Let ๐ด โ ๐๐(โ) be skew-symmetric. Prove that all non-zero eigenvalues of ๐ด are pure imaginary complex numbers. (A complex number is pure imaginary if it has the form ๐๐ for some nonzero ๐ โ โ.)
Suppose that a matrix ๐ด โ ๐๐(โ) has the property that all its eigenvalues are real and positive. Does it follow that vแต๐ด๐ฃ > 0 for all non-zero ๐ฃ โ โโฟ
Let ๐ด โ ๐๐(โ). Prove that ๐ด can be written in a unique way as ๐ด1 + ๐ด2 where ๐ด1 is symmetric and ๐ด2 is skew-symmetric. Prove that ๐ฃแต๐ด๐ฃ โฅ 0 for all ๐ฃ โ โ๐ if and only if the symmetric matrix ๐ด1 is positive definite.
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