[線代] 內積矩陣?!
Let V is an inner product space over R with dim(V) = 2
give B = {w1,w2} a basis of V
Denote E = <w1,w1> , F = <w1,w2>=<w2,w1> , G = <w2,w2>
[E F]
Then A = [ ] have positive determinant
[F G]
i.e. ∥w1∥^2 ∥w2∥^2 - <w1,w2>^2 > 0
This can be proved by Cauchy-Schwarz inequality
(且因為w1,w2線性獨立,所以等號不成立)
接著是我猜的,想請問一下對錯及證法：
Let V is an inner product space over R with dim(V) = n
give B = {w1,...,wn} a basis of V
Denote a_11 = <w1,w1> , ... a_nn = <wn,wn>
a_12 = <w1,w2> = <w2,w1> = a_21
if i=/=j , a_ij = <wi,wj> = <wj,wi>
[a_11 ‧‧‧ a_1n]
[ ‧ ‧‧‧ ‧ ]
Then A =[ ‧ ‧‧‧ ‧ ] has positive determinant
[ ‧ ‧‧‧ ‧ ]
[a_n1 ‧‧‧ a_nn]
如果這個結果是錯的話,那能否放寬,只要det(A) =/= 0 即可呢??
謝謝指教!!!
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