Consider a 3×3 matrix
Which statements are correct?
I、II、IV
I、IV、V
II、IV、V
I、II、III、V
I、V
Which statements are always correct?
I、IV、V
II、III、IV
III、IV、V
I、II、V
I、II、III
Given n×n matrices A and B , and there is an n×n invertible matrix P such
that B = P−1AP .
[I] A and B have the same trace
[II] A and B have the same eigenvectors
[III] A and B have the same determinant
[IV] A and B have the same eigenvalues
[V] A and B are simultaneously diagonalizable
Among the above statements, which are not always true?
II、IV、V
I、IV、V
I、III、IV
II、V
II、III
Among above statements, which are not always true?
II、III
III、V
III、IV、V
III、IV
I、II
Given singular value decomposition of a matrix H ∈Cm×n as H =UΣVH , where
U and V are m×m and n×n unitary matrices, Σ is an m×n diagonal
matrix composed of nonnegative singular values σ1,?,σT , 0,?, 0, where
r = rank(H) .
(1) Find the eigenvalues of HHH .
(2) Prove that
Find the LU decomposition of the following matrix:
Please use LU-decomposition to solve the following system of linear equations:
(1) Write X in term of the vector in B2 .
(2) Find the transformation matrix that converts a vector from in terms of Base B1 to
Base B2 .
Consider the vector space R3 with Euclidean inner product. Apply the
Gram-Schmidt process to transform the basis vector u1 = (0,1,0) , u2 = (1,1,1) and
u3 = (1,1,2) into an orthogonal basis {v1, v2 , v3} ; then normalize the orthogonal
basis vectors to obtain an orthonomal basis {q1, q2 , q3} .
Consider the vector space P3 of polynomials of degree less than 3, and the ordered
basis B = {x2 , x, 1} for P3 . Let T : P3 → P3 be the linear transformation such that
T(ax2 + bx + c) = (a − c)x2 − bx + 2c
Find the eigenvalues and the eigenvectors for the linear transformation T .
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