How do I find a vector x such that $x'Ax < 0 $ or make the matrix $A $ no longer SPD?











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I'm given a matrix a 3x3 matrix $A $ , where I need to find a vector x such that $x'Ax $ it gives me a negative number (or x'Ax < 0). I was thinking I have a vector x set to x = [a, b, c] and perform gaussian elimination but not sure where to go from there.



Any tips on how to approach this would be great!










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    I'm given a matrix a 3x3 matrix $A $ , where I need to find a vector x such that $x'Ax $ it gives me a negative number (or x'Ax < 0). I was thinking I have a vector x set to x = [a, b, c] and perform gaussian elimination but not sure where to go from there.



    Any tips on how to approach this would be great!










    share|cite|improve this question


























      up vote
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      down vote

      favorite









      up vote
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      down vote

      favorite











      I'm given a matrix a 3x3 matrix $A $ , where I need to find a vector x such that $x'Ax $ it gives me a negative number (or x'Ax < 0). I was thinking I have a vector x set to x = [a, b, c] and perform gaussian elimination but not sure where to go from there.



      Any tips on how to approach this would be great!










      share|cite|improve this question















      I'm given a matrix a 3x3 matrix $A $ , where I need to find a vector x such that $x'Ax $ it gives me a negative number (or x'Ax < 0). I was thinking I have a vector x set to x = [a, b, c] and perform gaussian elimination but not sure where to go from there.



      Any tips on how to approach this would be great!







      numerical-methods numerical-linear-algebra






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      edited Nov 14 at 10:56









      dmtri

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      asked Nov 14 at 10:44









      user208691

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          First of all, find the eigenvalues $lambda_k$ and eigenvectors $V_k$ associated with them. Two cases :




          • (a) either all $lambda_k$s are $geq 0$ , in such a case, you will never find an $X$ such that $X^TAX <0$.


          • (b) or one at least of the eigenvalues, say $lambda_1$ is $<0$, in this case, take $X=V_1$. Indeed, $X^TAX=V_1^T(AV_1)=V_1^T(lambda_1 V_1)$ (by definition of an eigenvector), which gives $lambda_1 |V_1|^2 < 0$. That's all.



          Considering the alternative you give in your title (find a way to make $A$ "no longer Symmetric Positive Definite"), you just have to "shift left" the spectrum of $A$. For example, if Spec$(A)={2,3,7}$, replace $A$ by, say, $A-2.001 I_3$, the spectrum of the new matrix will be shifted by $-2$ , i.e., will become${-0.001,0.999,4.999}$, and we are now in case (b)






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            First of all, find the eigenvalues $lambda_k$ and eigenvectors $V_k$ associated with them. Two cases :




            • (a) either all $lambda_k$s are $geq 0$ , in such a case, you will never find an $X$ such that $X^TAX <0$.


            • (b) or one at least of the eigenvalues, say $lambda_1$ is $<0$, in this case, take $X=V_1$. Indeed, $X^TAX=V_1^T(AV_1)=V_1^T(lambda_1 V_1)$ (by definition of an eigenvector), which gives $lambda_1 |V_1|^2 < 0$. That's all.



            Considering the alternative you give in your title (find a way to make $A$ "no longer Symmetric Positive Definite"), you just have to "shift left" the spectrum of $A$. For example, if Spec$(A)={2,3,7}$, replace $A$ by, say, $A-2.001 I_3$, the spectrum of the new matrix will be shifted by $-2$ , i.e., will become${-0.001,0.999,4.999}$, and we are now in case (b)






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              First of all, find the eigenvalues $lambda_k$ and eigenvectors $V_k$ associated with them. Two cases :




              • (a) either all $lambda_k$s are $geq 0$ , in such a case, you will never find an $X$ such that $X^TAX <0$.


              • (b) or one at least of the eigenvalues, say $lambda_1$ is $<0$, in this case, take $X=V_1$. Indeed, $X^TAX=V_1^T(AV_1)=V_1^T(lambda_1 V_1)$ (by definition of an eigenvector), which gives $lambda_1 |V_1|^2 < 0$. That's all.



              Considering the alternative you give in your title (find a way to make $A$ "no longer Symmetric Positive Definite"), you just have to "shift left" the spectrum of $A$. For example, if Spec$(A)={2,3,7}$, replace $A$ by, say, $A-2.001 I_3$, the spectrum of the new matrix will be shifted by $-2$ , i.e., will become${-0.001,0.999,4.999}$, and we are now in case (b)






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                up vote
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                First of all, find the eigenvalues $lambda_k$ and eigenvectors $V_k$ associated with them. Two cases :




                • (a) either all $lambda_k$s are $geq 0$ , in such a case, you will never find an $X$ such that $X^TAX <0$.


                • (b) or one at least of the eigenvalues, say $lambda_1$ is $<0$, in this case, take $X=V_1$. Indeed, $X^TAX=V_1^T(AV_1)=V_1^T(lambda_1 V_1)$ (by definition of an eigenvector), which gives $lambda_1 |V_1|^2 < 0$. That's all.



                Considering the alternative you give in your title (find a way to make $A$ "no longer Symmetric Positive Definite"), you just have to "shift left" the spectrum of $A$. For example, if Spec$(A)={2,3,7}$, replace $A$ by, say, $A-2.001 I_3$, the spectrum of the new matrix will be shifted by $-2$ , i.e., will become${-0.001,0.999,4.999}$, and we are now in case (b)






                share|cite|improve this answer














                First of all, find the eigenvalues $lambda_k$ and eigenvectors $V_k$ associated with them. Two cases :




                • (a) either all $lambda_k$s are $geq 0$ , in such a case, you will never find an $X$ such that $X^TAX <0$.


                • (b) or one at least of the eigenvalues, say $lambda_1$ is $<0$, in this case, take $X=V_1$. Indeed, $X^TAX=V_1^T(AV_1)=V_1^T(lambda_1 V_1)$ (by definition of an eigenvector), which gives $lambda_1 |V_1|^2 < 0$. That's all.



                Considering the alternative you give in your title (find a way to make $A$ "no longer Symmetric Positive Definite"), you just have to "shift left" the spectrum of $A$. For example, if Spec$(A)={2,3,7}$, replace $A$ by, say, $A-2.001 I_3$, the spectrum of the new matrix will be shifted by $-2$ , i.e., will become${-0.001,0.999,4.999}$, and we are now in case (b)







                share|cite|improve this answer














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                edited Nov 18 at 7:04

























                answered Nov 17 at 22:18









                Jean Marie

                28.2k41848




                28.2k41848






























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