You can do all of these in terms of matmul to some extent: Solving AX=B can be done with Newton's method to invert A, which boils down to matmuls. Matrix expo

You can do all of these in terms of matmul to some extent: Solving AX=B can be done with Newton's method to invert A, which boils down to matmuls. Matrix exponential is normally done with matmuls- the scale down, Taylor/Pade and square approach. Why do you need Cholesky? It's typically a means to an end, and when matmul is your primitive, you reach for it much less often. Eigendecomposition is hard. If we limit ourselves to symmetric, we could use a blocked Jacobi algorithm where we run a non-matmul Jacobi to do 128x128 off-diagonal blocks and then use the matmul unit to apply to the whole matrix- for large enough matrices, still bottlenecked on matmul. SVD we can get from Polar decomposition, which has purely-matmul iterations, and symmetric eigendecomposition. One does have to watch out for numerical stability and precision very carefully when doing all these!