## Partial derivatives

Let be a smooth map, and define the partial derivative as for each . Then is also a smooth map. The derivative with respect to is

where the represents the element of the matrix cofactors. In particular

.

Now let’s consider the second derivative for a special case that is affine with respect to , that is, is independent of :

– – – – – – – – – – – – , where

, and

– – – – – – .

I do not know if is zero (if it is, then the minus function will be convex).

——

In linear algebra, a symmetric matrix is said to be positive definite if is positive, for any non-zero vector .

Proposition. A symmetric , and a symmetric and positive-definite matrix can be simultaneously diagonalized: that is, there exists such that and .

Proposition. Suppose are symmetric matrices with and . Then.

Proof. Since is symmetric, there exists an orthogonal matrix with . Clearly and . So the -entries and . Therefore and .

Proposition. Suppose . If , then .

Proof. Note that . Then . Combining with we get .

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