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對稱矩陣

對稱矩陣. 轉置矩陣定義 (transpose matrix) 若 A 為任意 m × n 矩陣，則 A 的轉置矩陣為一 n × m 矩陣且以符號 A T 表示，其中 A T 的元素係將 A 的列與行交換。 ( A T ) ij =( A ) ji. 性質：設下列矩陣之階數使所有運算皆有意義，其中 k 為任義實數，則 (1) ( A ± B ) T = A T ± B T (2) ( kA ) T = k A T (3) ( AB ) T = B T A T (4) ( A T ) T = A. 對稱矩陣定義 (symmetric matrix)

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對稱矩陣

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  1. 對稱矩陣

  2. 轉置矩陣定義(transpose matrix) 若A為任意m×n矩陣，則A的轉置矩陣為一n×m矩陣且以符號AT表示，其中AT的元素係將A的列與行交換。 (AT)ij=(A)ji

  3. 性質：設下列矩陣之階數使所有運算皆有意義，其中k為任義實數，則性質：設下列矩陣之階數使所有運算皆有意義，其中k為任義實數，則 (1) (A±B)T=AT±BT (2) (kA)T=k AT (3) (AB)T= BT AT (4) (AT)T=A

  4. 對稱矩陣定義(symmetric matrix) 若方陣A滿足AT =A，則方陣A稱為對稱方陣。

  5. 例：若矩陣 試求AAT與ATA？ 解：

  6. 跡數定義(trace) 若A為一n階方陣，則方陣A主對角線元素的合稱為跡數(trace)，以符號tr(A)表示，即 tr(A)=a11+a22+…+ann 例：若矩陣 則 tr(A)=-1+5+7+0=11

  7. 性質：設下列矩陣之階數使所有運算皆有意義，其中k為任義實數，則性質：設下列矩陣之階數使所有運算皆有意義，其中k為任義實數，則 (1) tr(A±B)=tr(A)±tr(B) (2) tr(kA)=ktr(A) (3) tr(AB)=tr(BA) (4) tr(AT)=tr(A)

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