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Gram Matrices and Kernel Matrices
Gram Matrices and Kernel Matrices
3 questions
Difficulty 3-4
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foundation (3/10)
state theorem
The Gram matrix
G
ij
=
⟨
x
i
,
x
j
⟩
of a set of vectors
{
x
1
,
…
,
x
n
}
in an inner product space is always PSD. Why?
Hide and think first
A.
Gram matrices are always diagonal, and diagonal matrices with non-negative entries are PSD
B.
PSDness of Gram matrices is an axiom of inner product spaces, not a theorem
C.
For any coefficient vector
c
,
c
T
G
c
=
∥
∑
i
c
i
x
i
∥
2
≥
0
, directly from linearity of the inner product
D.
Gram matrices are always symmetric, and symmetric matrices are always PSD
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