Continuity of the determinant function | Physics Forums

Theorem: Let be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. InsightsBlog Forums HotThreads FeaturedThreads Login Register What'snew Search GeneralMath Calculus DifferentialEquations TopologyandAnalysis LinearandAbstractAlgebra DifferentialGeometry SetTheory,Logic,Probability,Statistics MATLAB,Maple,Mathematica,LaTeX Menu Login Register Navigation Installtheapp Install Moreoptions Contactus CloseMenu Youareusinganoutofdatebrowser.Itmaynotdisplaythisorotherwebsitescorrectly.Youshouldupgradeoruseanalternativebrowser. Forums Mathematics TopologyandAnalysis Continuityofthedeterminantfunction I Threadstarter dextercioby Startdate Sep26,2017 Tags continuous matrices topology dextercioby ScienceAdvisor HomeworkHelper InsightsAuthor 13,194 764 ThisissomethingIseekaproofof. Theorem:Let##\mbox{det}:\mbox{Mat}_{n\timesn}(\mathbb{R})\rightarrow\mathbb{R}##bethedeterminantfunctionassignedtoageneralnxnmatrixwithrealentries.Provethismappingiscontinuous. Myattempt.Continuitymustbejudgedin##\mathbb{R}##,soitshouldeventuallygodowntoanepsilon-deltaproof.Iwasfirstthinkingtoidentify##\mbox{Mat}_{n\timesn}(\mathbb{R})=\mathbb{R}^{n^2}##.HowdoIgofurther?   AnswersandReplies lavinia ScienceAdvisor GoldMember 3,275 663 dexterciobysaid: ThisissomethingIseekaproofof. Theorem:Let##\mbox{det}:\mbox{Mat}_{n\timesn}(\mathbb{R})\rightarrow\mathbb{R}##bethedeterminantfunctionassignedtoageneralnxnmatrixwithrealentries.Provethismappingiscontinuous. Myattempt.Continuitymustbejudgedin##\mathbb{R}##,soitshouldeventuallygodowntoanepsilon-deltaproof.Iwasfirstthinkingtoidentify##\mbox{Mat}_{n\timesn}(\mathbb{R})=\mathbb{R}^{n^2}##.HowdoIgofurther? Notethatthedeterminantisapolynomialinthe##n^2##coordinatesof##R^{n^2}##.Forinstancefora##R^4##itis##xw-yz##.   LaTeXGuide|BBcodeGuide Postreply Insertquotes… Share: Share RelatedThreadsonContinuityofthedeterminantfunction GraphsofContinuousFunctionsandtheSubspaceTopology LastPost Mar22,2013 Replies 1 Views 2K Continuousmappingsintopology. LastPost Sep20,2012 Replies 9 Views 2K M I Boundednessandcontinuousfunctions LastPost Jul16,2016 Replies 10 Views 2K Whatistheintuitivemeaningofcontinuityintopology? LastPost Sep1,2015 Replies 19 Views 4K L Simpletopologyprobleminvolvingcontinuity LastPost Mar21,2014 Replies 3 Views 2K Continuity,giventopologies. LastPost May29,2011 Replies 10 Views 3K S I Continuityofaninverseofafunction LastPost Aug7,2020 Replies 1 Views 742 I DefinitionsofContinuityinTopologicalSpaces... LastPost Mar7,2020 Replies 2 Views 1K I Understandingthedefinitionofcontinuousfunctions LastPost Dec7,2018 Replies 9 Views 1K W Continuityofthemaximum LastPost Jul15,2013 Replies 8 Views 2K B Forums Mathematics TopologyandAnalysis HotThreads I Duistermaat&Kolk...VolII...ProofofProposition6.1.2 I LookingforSomeoneStronginAnalysis I Fouriertransformofabeat I InEuclidianspace,closedballisequaltoclosureofopenball I ProofofInductionPrinciplestartingfromN,notfrom1 RecentInsights Insights TheHistoryandImportanceoftheRiemannHypothesis Insights SymmetryArgumentsandtheInfiniteWirewithaCurrent Insights ParallelProgrammingonaCPUwithAVX-512 Top