6) A matrix 𝐴 in 𝑀𝑛(ℝ) is called skew-symmetric if 𝐴...

6)
A matrix 𝐴 in 𝑀𝑛(ℝ) is called skew-symmetric if 𝐴ᵀ = −𝐴.
Show that the set of all skew-symmetric matrices in 𝑀𝑛(ℝ) is a subspace of 𝑀𝑛(ℝ) and determine its dimension (in term of n ).

7)
Let 𝐴 ∈ 𝑀𝑛(ℝ). Prove that vᵀ𝐴𝑣 = 0 for 𝑣 ∈ ℝ𝑛 if and only if 𝐴 is skew-symmetric.

8)
Let 𝐴 ∈ 𝑀𝑛(ℝ) be skew-symmetric. Prove that all non-zero eigenvalues of 𝐴 are pure imaginary complex numbers. (A complex number is pure imaginary if it has the form 𝑎𝑖 for some nonzero 𝑎 ∈ ℝ.)

9)
Suppose that a matrix 𝐴 ∈ 𝑀𝑛(ℝ) has the property that all its eigenvalues are real and positive. Does it follow that vᵀ𝐴𝑣 > 0 for all non-zero 𝑣 ∈ ℝⁿ

10)
Let 𝐴 ∈ 𝑀𝑛(ℝ). Prove that 𝐴 can be written in a unique way as 𝐴1 + 𝐴2 where 𝐴1 is symmetric and 𝐴2 is skew-symmetric. Prove that 𝑣ᵀ𝐴𝑣 ≥ 0 for all 𝑣 ∈ ℝ𝑛 if and only if the symmetric matrix 𝐴1 is positive definite.

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