課程名稱︰工程數學一
課程性質︰必修
課程教師︰劉進賢
開課學院：工學院
開課系所︰土木系
考試日期（年月日）︰2014/12/08
考試時限（分鐘）：120分鐘
試題 :
1.(30%)An upper triangular matrix is such a matrix that all the elements below
the main diagonal are zero.
(a) For an n ×n upper triangular matrix A =〔a_ij〕prove that the eigenvalues
of A are the entires on the main diagonal.
(b)Show that if λ is an eigenvalue of A that λ^k is also an eigenvlue of A^k.
Where k is a positive interger.
(c)Use the above results to find the eigenvalue of A^10 where
┌ ┐
│1 3 7 11│
A =│0 -1 3 8│
│0 0 -2 4│
│0 0 0 2│
└ ┘
2.(30%) Please judge that are the following matrices diagonalizable?Write the
reason or find a matrix P that diagonalize A and determine P^-1*A*P
(a) ┌ ┐ (b)┌ ┐ (c)┌ ┐
│-1 0 1│ │2 0 -2│ │-2 0 0 0│
A =│-1 3 0│ A =│0 3 0│ A =│ 0 -2 0 0│
│-4 13 -1│ │0 0 3│ │ 0 0 3 0│
└ ┘ └ ┘ │ 0 0 1 3│
└ ┘
3.(20%)Find the eigenspace and its dimension of the following symmetric
matrices.
(a) ┌ ┐ (b) ┌ ┐
│6 0 0│ │4 4 0 0│
A =│0 3 3│ A =│4 4 0 0│
│0 3 3│ │0 0 0 0│
└ ┘ │0 0 0 0│
└ ┘
4.(40%)Find a matrix P,which is orthonormal,that diagonalize the following
symmetric matrices.Determine P^-1*AP and compute A^20.
(a) ┌ ┐ (b)┌ ┐ (c) ┌ ┐ (d) ┌ ┐
│3 1│ │ 2 -1 -1│ │1 3 0 0│ │5 -2 0 0│
A =│1 3│ A =│-1 2 -1│ A =│3 1 0 0│ A =│-2 2 0 0│
└ ┘ │-1 -1 2│ │0 0 0 0│ │0 0 5 -2│
└ ┘ │0 0 0 0│ │0 0 -2 2│
└ ┘ └ ┘