[理工] [線代]兩題線代

[dkcheng (電磁霸主)]

1. Let V = M    be a vector space with the Frobenius inner product and W be a
            2x2


                                          [ 1 1 ]
subset of V defined by W = {A in V:trace (|     |A)=0}
                                          [ 1 0 ]

(1) Find a basis for the subspace of V consisting af all matrices that are

              [ 1 4 ]
orthogonal to |     |.
              [ 3 2 ]

(2) Is W a subspace of V ? If yes, find an orthonormal basis for W.

            [ 1 1 ]
(3) Let B = |     |. Find the orthogonal projection of B onto W. What is the
            [ 1 1 ]

distance from b to W ?



2. Let P be a 3x3 orthogonal projection matrix onto the plane 2x+2y-z=0

                         3
(1) What is the rank of P  ? What are its three eigenvalues ?

        3
(2) Is P  diagonalizable ? What are its three eigenvectors ?

            3                                        -1
(3) Let Q=2P -I . Is Q invertible ? If yes, what is Q   ?
               3



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推 louis719:1.只差在他是矩陣空間  我猜你應該是不會算矩陣內積??
→ louis719:2.P^3 = P 投影矩陣幾次方都還是投影矩陣 所以rankP^3=2
→ louis719:然後eigenvalue是0,1,1 所以可以對角化 兩個1對應到的就
→ louis719:是2x+2y+2z=0的解  0的eigenvector則是[2 2 1]
→ louis719:打錯[2 2 -1]才對
推 louis719:Q=2P^3-I = 2P - I 所以Q是reflection matrix
→ louis719:也就是對2x+2y-z=0這個平面做反射 所以Q=Q^-1
推 louis719:抱歉我推文第四句應該是  2x+2y-z=0的解