Category Archives: pow 二次型

證明 為正規矩陣 (normal matrix) 是一個充分條件。 Let and be normal matrices such that , where denotes the column space of . Prove that is a normal matrix. Note that is a normal matrix if .

任一正規矩陣 (normal matrix) 可表示為一個正規矩陣的平方。 Let be a normal matrix, i.e., . Prove that there exists a normal matrix such that .

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 .

證明反對稱矩陣的秩必為偶數。 Prove that the rank of a real skew-symmetric matrix is an even number.

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.

這是兩個實對稱矩陣以相合變換同時可對角化問題。 Let and be real symmetric matrices, and , . If there exists a such that is a positive semidefinite matrix and , then there exists a nonsingular matrix such that both and are diagonal. Note that denotes the nullspace … Continue reading →

若 是一個二階方陣且 ，證明存在一個么正 (unitary) 矩陣 使得 的主對角元為零。 Let be a matrix and . Show that there exists a unitary matrix such that the diagonal elements of are equal to zero.

給定正整數 ，證明任一矩陣 可分解為 。 Let be any complex matrix. Show that for each positive integer there exists a unique matrix such that .