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Is it enough to prove that the Kernel matrix is positive semidefinite to know that the function is a kernel? Or is it also necessary to prove that the matrix is symmetric?

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    $\begingroup$ Because "kernel" has so many distinct but related meanings, please give us some context or a clear definition of what you are trying to ask about. $\endgroup$
    – whuber
    Nov 10, 2023 at 15:40

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The kernel matrix will always be a symmetric (semi-) definite matrix because it gives the covariances of the images of the data points in the induced feature space, and covariance matrices must be symmetric.

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The question is a bit under-specified -- is it enough for what purpose?

The "kernel trick," which OP has included as a tag, is premised on Mercer kernels. Mercer's theorem considers symmetric PSD kernels.

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