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Matrix Functions

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1.Consider the 2 functions f1(t) and f2(t);
1. f1(t) = { a1.e^ -2t for t>=0 and
= { 0 for t<0 }
f2(t) = { a2.e^ -t + a3.e^ -2t for t>=0 and
= { 0 for t<0 }
Find a1,a2 and a3 such that f1(t) and f2(t) are orthonormal on the interval 0 to infinity.

2. If Q is an n x n symmetric matrix and a1,a2 are such that 0 < a1I <= Q <= a2I show that
(1/a2)I <= Q^ -1 <= (1/a1)I.

3.Suppose W(t) is an n x n matrix such that W(t) - aI is symmetric and positive semidefinite for all t, where a>0.Show there exists b>0 such tht det W(t) >= b for all t.

4.If A(t) is a continuously differentiable n x n matrix function that is invertible at each t, show that
d/dt A^ -1(t) = -A^ -1(t). dA(t)/dt. A^-1(t)

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Matrix functions are investigated. The solution is detailed and well presented.

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1. Proof:
and are orthogonal on , then we have
for any
Thus either or .
Therefore, the ...

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