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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?
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.
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.
Required, but never shown