Analysis II | Part IB, 2003

Let VV be the space of n×nn \times n real matrices. Show that the function

N(A)=sup{Ax:xRn,x=1}N(A)=\sup \left\{\|A \mathbf{x}\|: \mathbf{x} \in \mathbb{R}^{n},\|\mathbf{x}\|=1\right\}

(where \|-\| denotes the usual Euclidean norm on Rn\mathbb{R}^{n} ) defines a norm on VV. Show also that this norm satisfies N(AB)N(A)N(B)N(A B) \leqslant N(A) N(B) for all AA and BB, and that if N(A)<ϵN(A)<\epsilon then all entries of AA have absolute value less than ϵ\epsilon. Deduce that any function f:VRf: V \rightarrow \mathbb{R} such that f(A)f(A) is a polynomial in the entries of AA is continuously differentiable.

Now let d:VRd: V \rightarrow \mathbb{R} be the mapping sending a matrix to its determinant. By considering d(I+H)d(I+H) as a polynomial in the entries of HH, show that the derivative d(I)d^{\prime}(I) is the function HtrHH \mapsto \operatorname{tr} H. Deduce that, for any A,d(A)A, d^{\prime}(A) is the mapping Htr((adjA)H)H \mapsto \operatorname{tr}((\operatorname{adj} A) H), where adjA\operatorname{adj} A is the adjugate of AA, i.e. the matrix of its cofactors.

[Hint: consider first the case when AA is invertible. You may assume the results that the set UU of invertible matrices is open in VV and that its closure is the whole of VV, and the identity (adjA)A=detA.I(\operatorname{adj} A) A=\operatorname{det} A . I.]

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