Suppose A^T = -A (real antisymmetric matrix). Explain these facts about A: (a) x^TAx = 0 for every real vector x. (b) The eigenvalues of A are pure imaginary. (c) The determinant of A is positive or zero (not negative). For (a), multiply out an example of x^TAx and watch terms cancel. Or reverse x^T

Question

Suppose {A}^{T} = -A (real antisymmetric matrix). Explain these facts about A: (a) {x}^{T}Ax = 0 for every real vector x. (b) The eigenvalues of A are pure imaginary. (c) The determinant of A is positive or zero (not negative). For (a), multiply out an example of {x}^{T}Ax and watch terms cancel. Or reverse {x}^{T}(Ax) to (Ax)^{T}x . For (b), Az = \lambda z leads to \bar{z}^{T} Az = \lambda \bar{z}^{T}z = \lambda \left\|z \right\|^{2} . Part(a) shows that \bar{z}^{T} Az = (x - i y )^{T} A (x + i y) has zero real part. Then (b) helps with (c).

Accepted Answer

(a) [latex]{x}^{T}(Ax) = (Ax)^{T}x = {x}^{T}{A}^{T}x = -{x}^{T}Ax[/latex].

(b) [latex]\bar{z}^{T}Az[/latex] is pure imaginary, its real part is [latex]{x}^{T}Ax + {y}^{T}Ay = 0 + 0[/latex]

(c) [latex]det A = {\lambda}_{1} \cdots {\lambda}_{n} \ge 0[/latex]: pairs of [latex]\lambda[/latex] ’s = ib,-ib.

Question 6.4.31: Suppose A^T = -A (real antisymmetric matrix). Explain these ...

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