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Exercise 6.3.5

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1.

Just write it down just as that in the proof of (c).

(a)

Compute

L_{{(A + B)}^{∗}} = (L_{A + B}^{∗} = {(L_{A} + L_{B})}^{∗}

= {(L_{A})}^{∗} + {(L_{B})}^{∗} = L_{A^{∗}} + L_{B^{∗}} = L_{A^{∗} + B^{∗}} .

(b)

Compute

L_{{(cA)}^{∗}} = {(L_{cA})}^{∗} = {(c L_{A})}^{∗}

= \bar{c} {(L_{A})}^{∗} = \bar{c} L_{A^{∗}} = L_{\bar{c} A^{∗}} .

(d)

Compute

L_{A^{∗∗}} = {(L_{A})}^{∗∗} = L_{A} .

(e)

Compute

L_{I^{∗}} = {(L_{I})}^{∗} = L_{I} .

2.

The statement for nonsquare matrices has no difference with the Corollary but the statement (e) cannot holds since there’s no nonsquare identity matrix. To this result come from that

A^{∗}

is the conjugate of

A^{t}

jephianlin

2011-06-27 00:00

Exercise 6.3.5

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