For an orthogonal matrix $Q$, prove $operatornamecond(Q)=1$Orthogonal matrix normWhy is the matrix product of 2 orthogonal matrices also an orthogonal matrix?Finding an orthogonal Matrix with distinct eigenvaluesOrthogonal Matrix Eigenvaluehow to prove this matrix is orthogonalShow that for $n times n$ orthogonal matrix $A$ that $operatornameCond(A)leq n$Can we prove that the eigenvalues of an $ntimes n$ orthogonal matrix are $pm 1$ by only using the definition of orthogonal matrix?Show non-singularity of orthogonal matrixProve that the multiples of two orthogonal eigenvectors with a matrix are also orthogonalProve that $-1$ is an eigenvalue of an orthogonal matrix $A in M_4 times 4 (Bbb R)$ with $det(A)=-1$Band matrix conjugation relation between orthogonal matrices

What is the purpose of a disclaimer like "this is not legal advice"?

-1 to the power of a irrational number

Smooth vector fields on a surface modulo diffeomorphisms

Idiom for feeling after taking risk and someone else being rewarded

Can one live in the U.S. and not use a credit card?

Do black holes violate the conservation of mass?

Use Mercury as quenching liquid for swords?

Finding the minimum value of a function without using Calculus

Is it appropriate to ask a former professor to order a book for me through an inter-library loan?

Either of .... (Plural/Singular)

PTIJ: Sport in the Torah

The (Easy) Road to Code

Can I negotiate a patent idea for a raise, under French law?

Is this Paypal Github SDK reference really a dangerous site?

How do spaceships determine each other's mass in space?

Trocar background-image com delay via jQuery

Automaton recognizing ambiguously accepted words of another automaton

Create chunks from an array

If sound is a longitudinal wave, why can we hear it if our ears aren't aligned with the propagation direction?

Is there a logarithm base for which the logarithm becomes an identity function?

Volume of hyperbola revolved about the y -axis

How can I portion out frozen cookie dough?

What is Tony Stark injecting into himself in Iron Man 3?

Insult for someone who "doesn't know anything"



For an orthogonal matrix $Q$, prove $operatornamecond(Q)=1$


Orthogonal matrix normWhy is the matrix product of 2 orthogonal matrices also an orthogonal matrix?Finding an orthogonal Matrix with distinct eigenvaluesOrthogonal Matrix Eigenvaluehow to prove this matrix is orthogonalShow that for $n times n$ orthogonal matrix $A$ that $operatornameCond(A)leq n$Can we prove that the eigenvalues of an $ntimes n$ orthogonal matrix are $pm 1$ by only using the definition of orthogonal matrix?Show non-singularity of orthogonal matrixProve that the multiples of two orthogonal eigenvectors with a matrix are also orthogonalProve that $-1$ is an eigenvalue of an orthogonal matrix $A in M_4 times 4 (Bbb R)$ with $det(A)=-1$Band matrix conjugation relation between orthogonal matrices













0












$begingroup$



Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$




I succeed to solve it with eigenvalues but I'm looking for an easier way.






matrices orthogonal-matrices condition-number spectral-norm







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
    $endgroup$
    – Minus One-Twelfth
    yesterday
















0












$begingroup$



Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$




I succeed to solve it with eigenvalues but I'm looking for an easier way.






matrices orthogonal-matrices condition-number spectral-norm







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
    $endgroup$
    – Minus One-Twelfth
    yesterday














0












0








0





$begingroup$



Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$




I succeed to solve it with eigenvalues but I'm looking for an easier way.






matrices orthogonal-matrices condition-number spectral-norm







share|cite|improve this question











$endgroup$





Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$




I succeed to solve it with eigenvalues but I'm looking for an easier way.






matrices orthogonal-matrices condition-number spectral-norm




matrices orthogonal-matrices condition-number spectral-norm






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









Rodrigo de Azevedo

13.1k41960




13.1k41960










asked yesterday









J. DoeJ. Doe

16811




16811







  • 1




    $begingroup$
    An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
    $endgroup$
    – Minus One-Twelfth
    yesterday













  • 1




    $begingroup$
    An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
    $endgroup$
    – Minus One-Twelfth
    yesterday








1




1




$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday





$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday











1 Answer
1






active

oldest

votes


















0












$begingroup$

An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.



So:
$$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
Since $Q^-1$ is orthogonal as well, the result follows.






share|cite|improve this answer









$endgroup$












    Your Answer





    StackExchange.ifUsing("editor", function ()
    return StackExchange.using("mathjaxEditing", function ()
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    );
    );
    , "mathjax-editing");

    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "69"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader:
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    ,
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3140446%2ffor-an-orthogonal-matrix-q-prove-operatornamecondq-1%23new-answer', 'question_page');

    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.



    So:
    $$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
    Since $Q^-1$ is orthogonal as well, the result follows.






    share|cite|improve this answer









    $endgroup$

















      0












      $begingroup$

      An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.



      So:
      $$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
      Since $Q^-1$ is orthogonal as well, the result follows.






      share|cite|improve this answer









      $endgroup$















        0












        0








        0





        $begingroup$

        An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.



        So:
        $$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
        Since $Q^-1$ is orthogonal as well, the result follows.






        share|cite|improve this answer









        $endgroup$



        An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.



        So:
        $$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
        Since $Q^-1$ is orthogonal as well, the result follows.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        I like SerenaI like Serena

        4,2721722




        4,2721722



























            draft saved

            draft discarded
















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …


            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.

            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3140446%2ffor-an-orthogonal-matrix-q-prove-operatornamecondq-1%23new-answer', 'question_page');

            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Solar Wings Breeze Design and development Specifications (Breeze) References Navigation menu1368-485X"Hang glider: Breeze (Solar Wings)"e

            Ace MagicMainair BladeMainair RapierP&M ExplorerP&M GT450P&M PulsRPegasus QuantumPegasus QuikSolar Wings AceSolar Wings BreezeSolar Wings FeverSolar Wings RumourSolar Wings RushSolar Wings ScandalSolar Wings StormSolar Wings TyphoonSolar Wings Whisper Solar Wings aircraftHang gliders Britishhigh-winghang gliderSolar WingsManton, WiltshirealuminumDacroncable bracedkingpostaspect ratio Solar Wings Breeze Role Hang glider National origin United Kingdom Manufacturer Solar Wings Introduction 1996 Status Production completed Unit cost £2,038 (2004) The Solar Wings Breeze is a British high-wing, single-place, hang glider that was designed and produced by Solar Wings of Manton, Wiltshire starting in 1996. Now out of production, when it was available the aircraft was supplied complete and ready-to-fly. [1] [2] Design and development The Breeze was designed as an easy to fly intermediate glider. It is made from aluminum tubing, with the double-surface wing covered in Dacron s...
            Read more

            Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee

            Image
            Parai AttamNatarajaTandavaRasa lilaLasyaBollywood song and danceHindi dance songsNautchAndhra PradeshAssamHimachal PradeshOdishaPunjabTamil NaduMizoramKerala KalamandalamKerala Lalitakala AcademyKerala Sangeetha Nadaka Academy, ThrissurKerala Folklore AcademyKshetram vadyamKuzhal PattuMappila PaattuPanchari melamPanchavadyamPandi MelamParisha VadyamPulluvan PaattuSopanamThayambakaKerala mural paintingKalaripayattuSarpa KavuTharavadTemplatesPortalCategoryWikiProject KathakaliTheatre in IndiaArts of KeralaClassical theatre of india Malayalamclassical Indian danceMalayalamKeralaCEHindu templesPuranasNatya ShastraNatya ShastraBharataShivaMalayalamKeralaShivaKrishnaKrishnaJayadevaoperaticKrishnaVishnuRamaShivaSuryaYudhishthiraArjunaNalaRavanaDushasanaHiranyakashipuSitaHanumanGarudaHamsaGuṇaSamkhyaHindu philosophysign languagemudrasUnnayi VariyarplaywrightsNalaDamayantimusical modeChristianityMiguel de CervantesJohann Wolfgang von GoetheWilliam ShakespeareGurukulTravancorePalakkadBrahmi...
            Read more

            Method to test if a number is a perfect power? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Detecting perfect squares faster than by extracting square rooteffective way to get the integer sequence A181392 from oeisA rarely mentioned fact about perfect powersHow many numbers such $n$ are there that $n<100,lfloorsqrtn rfloor mid n$Check perfect squareness by modulo division against multiple basesFor what pair of integers $(a,b)$ is $3^a + 7^b$ a perfect square.Do there exist any positive integers $n$ such that $lfloore^nrfloor$ is a perfect power? What is the probability that one exists?finding perfect power factors of an integerProve that the sequence contains a perfect square for any natural number $m $ in the domain of $f$ .Counting Perfect Powers

            Image
            Proof of work - 51% attack Would it be possible to dictate a bech32 address as a list of English words? How do I find out the mythology and history of my Fortress? Sentence order: Where to put もう Is there any word for a place full of confusion? Do wooden building fires get hotter than 600°C? What does it mean that physics no longer uses mechanical models to describe phenomena? Project Euler #1 in C++ How fail-safe is nr as stop bytes? Why does it sometimes sound good to play a grace note as a lead in to a note in a melody? How to draw/optimize this graph with tikz How much damage would a cupful of neutron star matter do to the Earth? What would you call this weird metallic apparatus that allows you to lift people? Importance of からだ in this sentence "Lost his faith in humanity in the trenches of Verdun" — last line of an SF story A term for a woman complaining about things/begging in a cute/childish way How to ...
            Read more