Recall that a real square matrix A is called skew symmetric if .
a. If A is skew symmetric, is skew symmetric as well? Or is symmetric?
b. If is skew symmetric, what can you say about the definiteness of ? What about the eigenvalues of ?
c. What can you say about the complex eigenvalues of a skew-symmetric matrix? Which skew-symmetric matrices are diagonalizable over ?
Therefore the solution is
a. is symmetric.
b. is a negative definite.
c. The zero matrix is the only skew-symmetric matrix that is diagonalizable over
a.) A symmetric matrix is equal to its transpose
and a skew symmetric matrix is a matrix whose transpose is equal to its negative
Now, if is a skew symmetric matrix, then as we just mentioned: , now to check if would be skew symmetric as well, we have:
therefore, is not skew symmetric, it is just symmetric.
b.) We have
from part (a) we mentioned A that is said to be a skew matrix if , so
therefore, for all , which implies that is a negative definite (its eigenvalues are less than or equal to 0).
c.) Let be an eigenvector corresponding to the eigenvalue,
multiply both sides by
note: the bar" -" above v is for complex conjugation.
for the left-hand side suppose we have column vectors of the same size, a and b, then is a matrix which we can think of as a scalar. Now, taking transposes, since is , it is its own transpose, so
here let and localid="1659623457872" , then for our left-hand side
since A is skew-symmetric, we have . Substituting,
taking the conjugate of yields , notice we can substitute this into the left-hand side above
which implies and . We also conclude that the zero matrix is the only skew-symmetric matrix that is diagonalizable over .
If you sell two cows and five sheep and you buy pigs, you gain coins. If you sell three cows and three pigs and buy nine sheep, you break even. If you sell six sheep and eight pigs and you buy five cows, you lose coins. What is the price of a cow, a sheep, and a pig, respectively? (Nine Chapters, Chapter 8, Problem 8)
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