Tag Archives: 正定矩陣

利用相合 (congruence) 變換證明若 ，則 。 Let and be Hermitian matrices. We will write that if is positive definite. The inequality means that is positive definite. Prove that if , then .

證明正定矩陣的伴隨矩陣 (adjugate) 也是一個正定矩陣。 Prove that if is a real symmetric positive definite then is also a symmetric positive definite matrix.

若可對角化矩陣的特徵值皆為實數，則此矩陣可表示為正定矩陣與 Hermitian 矩陣之積。 Let be a diagonalizable matrix with real eigenvalues. Show that can be represented as , where is a positive definite matrix and is a Hermitian matrix.

這是關於二次型的性質與判別問題，取自“台聯大2014年碩士班招生考試試題 (電機類工程數學D)”。 Let be an real matrix. Which of the following statements are true? (a) If all the eigenvalues of are positive, then for every nonzero . (b) If all the eigenvalues of are positive, then . (c) If for … Continue reading →