Understanding Determinants and Their Properties
Explore the basic properties of determinants in high dimensions, including how they relate to volume and area in different dimensions. Learn about exchanging rows, linearity, and other key concepts related to determinants. Check out illustrative examples and understand the significance of determinants in various mathematical contexts.
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Presentation Transcript
Basic Properties of Determinant Volume in high dimensions
Determinant in High School ?1 ?4 ?7 ?2 ?5 ?8 ?3 ?6 ?9 2 X 2 3 x 3 ? = ? =? ? ? ? ?7,?8,?9 |??? ? | ! (c,d) V (a,b) ?1,?2,?3 ?4,?5,?6
Three Basic Properties Basic Property 1: ??? ? = 1 Basic Property 2: Exchange rows only reverses the sign of det (do not change absolute value) Basic Property 3: Determinant is linear for each row Area in 2d and Volume in 3d have the above properties So det is Volume in high dimension?
Three Basic Properties Basic Property 1: ??? ? = 1 1 1 1 0 0 0 1 0 0 0 1 ?2=1 0 1 ?3= 0 ??? ?2 = 1 ??? ?3 = 1
Three Basic Properties ? ? ? ? ? ? ? ? Basic Property 2: Exchanging rows only reverses the sign of det 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 ??? = 1 1 0 0 1 ??? = 1 ??? = 1 0 1 1 0 ??? = 1 ??? = 1
Three Basic Properties Basic Property 2: Exchanging rows only reverses the sign of det If a matrix A has 2 equal rows ??? ? = 0 exchange the two rows ? ? ??? ? = ? = ??? ? = ? Exchanging the two equal rows yields the same matrix
Three Basic Properties Basic Property 3: Determinant is linear for each row 3-a ?? ? ?? ? ? ? ? ? ??? = ???? (c,d) (c,d) V V V (2a,2b) (a,b)
Three Basic Properties Basic Property 3: Determinant is linear for each row 3-a ?? ? ?? ? ? ? ? ? ??? = ???? 2? 2? 2? 2? ??? 2? ? ? = ??? Q: find ??? 2? ? ? 2? ? ? ? ? ? If A is n x n = 2??? = 4??? 2? A:??? 2? = 2???? ?
Three Basic Properties Basic Property 3: Determinant is linear for each row 3-a ?? ? ?? ? ? ? ? ? ??? = ???? A row of zeros Set ? = 0! ??? ? = 0 A row of zeros volume is zero
Three Basic Properties Basic Property 3: Determinant is linear for each row ? + ? ? + ? ? ? ? ? ? ? ? ? ? ? 3-b ??? = ??? + ??? (c,d) (c,d) (a+a ,b+b ) (a+a ,b+b ) (a,b) (a,b)
Three Basic Properties Basic Property 3: Determinant is linear for each row Subtract k x row i from row j (elementary row operation) Determinant doesn t change ? ? ??? ? ?? ? ?? ? ? ? ? ? ? 3-b = ??? + ??? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? 3-a = ??? ???? = ???
Formula from Three Properties 2 1 1 0 0 1 0 1 1 0 = 1 = 1 = det? 0 ? det? ? ? +det0 ? ? 3-b ? ? ? = det? 0 0+ det? 3-a 0 ? +det0 ? 0+ det0 3-a ? ? 3-b ? 0 3-a ? 0 3-a = ?? = bc = 0 = 0 = ?? ??
Finally, we get 3 x 3 x 3 matrices Most of them have zero determinants ?11 ?21 ?31 ?12 ?22 ?32 ?13 ?23 ?33 det ?11 ?21 ?31 0 0 0 ?12 ?22 ?32 0 0 0 ?13 ?23 ?33 ?22 ?32 ?23 ?33 ?21 ?31 ?23 ?33 ?21 ?31 ?22 ?32 = det +det +det ?11 0 ?31 0 0 0 ?11 ?21 ?31 0 0 0 0 ?11 0 ?31 0 0 0 ?22 ?32 +??? = ??? ?23 ?33 +??? ?33 ?32 ?33 ?32 ?11 ?21 ?31 0 0 0 0 0 0 ?11 ?21 0 0 0 0 0 0 ?11 ?21 0 0 0 0 0 0 = ??? +??? +??? ?32 ?33
?11 ?21 ?31 ?12 ?22 ?32 ?13 ?23 ?33 3! matrices have non-zero rows det 0 ?12 0 0 0 0 ?11 0 0 ?11?22?33 0 0 0 ?11 0 0 ?11?23?32 0 0 0 ?21 0 ?12?21?33 + ?22 0 = ?23 0 + ?33 ?33 ?32 0 0 ?12 0 0 0 0 0 0 ?13 0 0 0 0 0 ?13 0 0 ?23 0 + ?21 0 ?13?21?32 + ?22 0 + ?31 ?32 ?31 ?13?22?31 ?12?23?31 Pick an element at each row, but they can not be in the same column.
Formula from Three Properties Given an n x n matrix A ??? ? = ??? ?? ?! ????? ?1??2??3? ??? Format of each term: Find an element in each row permutation of 1,2, , n
Example 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 ??? 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 = ??? +??? +1 -1
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