6) A matrix ๐ด in ๐๐(โ) is called skew-symmetric if ๐ด...

Question

6)
A matrix ๐ด in ๐๐(โ) is called skew-symmetric if ๐ดแต = โ๐ด.
Show that the set of all skew-symmetric matrices in ๐๐(โ) is a subspace of ๐๐(โ) and determine its dimension (in term of n ).

7)
Let ๐ด โ ๐๐(โ). Prove that vแต๐ด๐ฃ = 0 for ๐ฃ โ โ๐ if and only if ๐ด is skew-symmetric.

8)
Let ๐ด โ ๐๐(โ) be skew-symmetric. Prove that all non-zero eigenvalues of ๐ด are pure imaginary complex numbers. (A complex number is pure imaginary if it has the form ๐๐ for some nonzero ๐ โ โ.)

9)
Suppose that a matrix ๐ด โ ๐๐(โ) has the property that all its eigenvalues are real and positive. Does it follow that vแต๐ด๐ฃ > 0 for all non-zero ๐ฃ โ โโฟ

10)
Let ๐ด โ ๐๐(โ). Prove that ๐ด can be written in a unique way as ๐ด1 + ๐ด2 where ๐ด1 is symmetric and ๐ด2 is skew-symmetric. Prove that ๐ฃแต๐ด๐ฃ โฅ 0 for all ๐ฃ โ โ๐ if and only if the symmetric matrix ๐ด1 is positive definite.

Solution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

By purchasing this solution you'll be able to access the following files:
Solution.pdf.

\$18.00
for this solution

or FREE if you
register a new account!

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available Linear Algebra Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.