6)
A matrix ð´ in ð‘€ð‘›(â„) is called skew-symmetric if ð´áµ€ = âˆ’ð´.
Show that the set of all skew-symmetric matrices in ð‘€ð‘›(â„) is a subspace of ð‘€ð‘›(â„) and determine its dimension (in term of n ).

7)
Let ð´ âˆˆ ð‘€ð‘›(â„). Prove that váµ€ð´ð‘£ = 0 for ð‘£ âˆˆ â„ð‘› if and only if ð´ is skew-symmetric.

8)
Let ð´ âˆˆ ð‘€ð‘›(â„) be skew-symmetric. Prove that all non-zero eigenvalues of ð´ are pure imaginary complex numbers. (A complex number is pure imaginary if it has the form ð‘Žð‘– for some nonzero ð‘Ž âˆˆ â„.)

9)
Suppose that a matrix ð´ âˆˆ ð‘€ð‘›(â„) has the property that all its eigenvalues are real and positive. Does it follow that váµ€ð´ð‘£ > 0 for all non-zero ð‘£ âˆˆ â„â¿

10)
Let ð´ âˆˆ ð‘€ð‘›(â„). Prove that ð´ can be written in a unique way as ð´1 + ð´2 where ð´1 is symmetric and ð´2 is skew-symmetric. Prove that ð‘£áµ€ð´ð‘£ â‰¥ 0 for all ð‘£ âˆˆ â„ð‘› if and only if the symmetric matrix ð´1 is positive definite.

6) A matrix ð´ in ð‘€ð‘›(â„) is called skew-symmetric if ð´...

6) A matrix ð´ in ð‘€ð‘›(â„) is called skew-symmetric if ð´...