110年 - 110 國立中央大學_碩士班招生考試_統計研究所/不分組(一般生、在職生):基礎數學#105322

科目：研究所、轉學考(插大)-基礎數學 | 年份：110年 | 選擇題數：0 | 申論題數：10

試卷資訊

所屬科目：研究所、轉學考(插大)-基礎數學

選擇題 (0)

申論題 (10)

(a) (10%) (Jensen's inequality) Verify that if

is a convex function

(b) (10%) Verify

(Hint: Use Jensen's inequality)

(a) (10%) Compute

(b) (10%) Show that
Hint: Show

(a) (10%) Find the eigenvalue for H². In addition, verify H² is positive semi-definite and trace(H²] =rank(H²]

(b) (10%) Find the eigenvalue for I - H. In addition, verify I -'H is positive semi-deinite and
trace[I - H] = rank[I -H]
where I is the identity matrix.

(a) (10%) Evaluate H²and obtain possible e to achieve H²= I.

(b) (10%) Evaluate HV and obtain possible O to achieve HV = -V.

5. (10%) Find the minimum distance from a point on the surfaces x+ y + z = c with some constant c to the origin.

6. (10%) Given the two by two matrix
Evaluate using the eigenvalue decomposition .Note that G⁰ = I where I is the identity matrix.