Transcription of Transpose & Dot Product - Stanford University
{{id}} {{{paragraph}}}
Example: bachelor of science
Transpose & Dot Product - Stanford University
Transpose & Dot ProductDef:Thetransposeof anm nmatrixAis then mmatrixATwhosecolumns are the rows : The columns ofATare the rows ofA. The rows ofATare the :IfA=[1 2 34 5 6], thenAT= 1 42 53 6 .Convention:From now on, vectorsv Rnwill be regarded as columns ( :n 1 matrices). Therefore,vTis a row vector (a 1 nmatrix).Observation:Letv,w Rn. ThenvTw=v w. This is because:vTw=[v1 vn] =v1w1+ +vnwn=v theory is concerned, the key property of transposes is the following:Prop :LetAbe anm nmatrix. Then forx Rnandy Rm:(Ax) y=x (ATy).Here, is the dot Product of ExampleLetAbe a 5 3 matrix, soA:R3 R5. N(A) is a subspace of C(A) is a subspace ofThe transposeATis amatrix, soAT: C(AT) is a subspace of N(AT) is a subspace ofObservation:BothC(AT) andN(A) are subspaces of. Might therebe a geometric relationship between the two? (No, they re not equal.) :BothN(AT) andC(A) are subspaces of.
Def: Let Q: Rn!R be a quadratic form. We say Qis positive de nite if Q(x) >0 for all x 6= 0. We say Qis negative de nite if Q(x) <0 for all x 6= 0. We say Qis inde nite if there are vectors x for which Q(x) >0, and also vectors x for which Q(x) <0. Def: Let Abe a symmetric matrix. We say Ais positive de nite if Q A(x) = xTAx >0 for all x 6= 0.
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: