How to classify the degenerate stationary points of a multivariate function?
I have a multivariate function in one of whose critical points the Hessian matrix is singular. Is there any general method to determine the type of this critical point? Would be worth plotting the function in the direction of the eigenvector associated to a zero eigenvalue of the Hessian matrix?
Suppose that there is a function $fleft( mathbf{x}_0 + mathbf{xi } right) approx fleft( {{mathbf{x}}_{0}} right)+frac{1}{2}{{mathbf{xi }}^{text{T}}}mathbf{H}mathbf{xi }$, where $mathbf{x}_0$ is a given stationary point, $mathbf{xi}=mathbf{x}-mathbf{x}_0$ is the vector of relative coordinates, and $mathbf{H}=mathbf{H}left( {{mathbf{x}}_{0}} right)$ is the Hessian matrix at $mathbf{x}=mathbf{x}_0$.
Then we can obtain the following form: $gleft( mathbf{zeta } right)=2left[ fleft( {{mathbf{z}}_{0}}+mathbf{zeta } right)-fleft( {{mathbf{z}}_{0}} right) right]={{mathbf{zeta }}^{text{T}}}text{diag}left( {{lambda }_{i}} right)mathbf{zeta }$, where ${{mathbf{z}}_{0}}={{mathbf{V}}^{text{T}}}{{mathbf{x}}_{0}}$ and $mathbf{zeta }={{mathbf{V}}^{text{T}}}mathbf{xi }$, $lambda_i$ is the $i$th eigenvalue of H, and V is the eigenvector matrix of H. What should I plot to see the behavior of the function around $mathbf{x}_0$?
multivariable-calculus hessian-matrix
asked Dec 10 '18 at 13:44
Roloka
566
add a comment |
I have a multivariate function in one of whose critical points the Hessian matrix is singular. Is there any general method to determine the type of this critical point? Would be worth plotting the function in the direction of the eigenvector associated to a zero eigenvalue of the Hessian matrix?
Suppose that there is a function $fleft( mathbf{x}_0 + mathbf{xi } right) approx fleft( {{mathbf{x}}_{0}} right)+frac{1}{2}{{mathbf{xi }}^{text{T}}}mathbf{H}mathbf{xi }$, where $mathbf{x}_0$ is a given stationary point, $mathbf{xi}=mathbf{x}-mathbf{x}_0$ is the vector of relative coordinates, and $mathbf{H}=mathbf{H}left( {{mathbf{x}}_{0}} right)$ is the Hessian matrix at $mathbf{x}=mathbf{x}_0$.
Then we can obtain the following form: $gleft( mathbf{zeta } right)=2left[ fleft( {{mathbf{z}}_{0}}+mathbf{zeta } right)-fleft( {{mathbf{z}}_{0}} right) right]={{mathbf{zeta }}^{text{T}}}text{diag}left( {{lambda }_{i}} right)mathbf{zeta }$, where ${{mathbf{z}}_{0}}={{mathbf{V}}^{text{T}}}{{mathbf{x}}_{0}}$ and $mathbf{zeta }={{mathbf{V}}^{text{T}}}mathbf{xi }$, $lambda_i$ is the $i$th eigenvalue of H, and V is the eigenvector matrix of H. What should I plot to see the behavior of the function around $mathbf{x}_0$?
multivariable-calculus hessian-matrix
asked Dec 10 '18 at 13:44
Roloka
566
add a comment |
I have a multivariate function in one of whose critical points the Hessian matrix is singular. Is there any general method to determine the type of this critical point? Would be worth plotting the function in the direction of the eigenvector associated to a zero eigenvalue of the Hessian matrix?
Suppose that there is a function $fleft( mathbf{x}_0 + mathbf{xi } right) approx fleft( {{mathbf{x}}_{0}} right)+frac{1}{2}{{mathbf{xi }}^{text{T}}}mathbf{H}mathbf{xi }$, where $mathbf{x}_0$ is a given stationary point, $mathbf{xi}=mathbf{x}-mathbf{x}_0$ is the vector of relative coordinates, and $mathbf{H}=mathbf{H}left( {{mathbf{x}}_{0}} right)$ is the Hessian matrix at $mathbf{x}=mathbf{x}_0$.
Then we can obtain the following form: $gleft( mathbf{zeta } right)=2left[ fleft( {{mathbf{z}}_{0}}+mathbf{zeta } right)-fleft( {{mathbf{z}}_{0}} right) right]={{mathbf{zeta }}^{text{T}}}text{diag}left( {{lambda }_{i}} right)mathbf{zeta }$, where ${{mathbf{z}}_{0}}={{mathbf{V}}^{text{T}}}{{mathbf{x}}_{0}}$ and $mathbf{zeta }={{mathbf{V}}^{text{T}}}mathbf{xi }$, $lambda_i$ is the $i$th eigenvalue of H, and V is the eigenvector matrix of H. What should I plot to see the behavior of the function around $mathbf{x}_0$?
multivariable-calculus hessian-matrix
asked Dec 10 '18 at 13:44
Roloka
566
I have a multivariate function in one of whose critical points the Hessian matrix is singular. Is there any general method to determine the type of this critical point? Would be worth plotting the function in the direction of the eigenvector associated to a zero eigenvalue of the Hessian matrix?
Suppose that there is a function $fleft( mathbf{x}_0 + mathbf{xi } right) approx fleft( {{mathbf{x}}_{0}} right)+frac{1}{2}{{mathbf{xi }}^{text{T}}}mathbf{H}mathbf{xi }$, where $mathbf{x}_0$ is a given stationary point, $mathbf{xi}=mathbf{x}-mathbf{x}_0$ is the vector of relative coordinates, and $mathbf{H}=mathbf{H}left( {{mathbf{x}}_{0}} right)$ is the Hessian matrix at $mathbf{x}=mathbf{x}_0$.
Then we can obtain the following form: $gleft( mathbf{zeta } right)=2left[ fleft( {{mathbf{z}}_{0}}+mathbf{zeta } right)-fleft( {{mathbf{z}}_{0}} right) right]={{mathbf{zeta }}^{text{T}}}text{diag}left( {{lambda }_{i}} right)mathbf{zeta }$, where ${{mathbf{z}}_{0}}={{mathbf{V}}^{text{T}}}{{mathbf{x}}_{0}}$ and $mathbf{zeta }={{mathbf{V}}^{text{T}}}mathbf{xi }$, $lambda_i$ is the $i$th eigenvalue of H, and V is the eigenvector matrix of H. What should I plot to see the behavior of the function around $mathbf{x}_0$?
multivariable-calculus hessian-matrix
multivariable-calculus hessian-matrix
asked Dec 10 '18 at 13:44
Roloka
566
asked Dec 10 '18 at 13:44
Roloka
566
edited Dec 11 '18 at 8:54
asked Dec 10 '18 at 13:44
Roloka
566
asked Dec 10 '18 at 13:44
Roloka
566
asked Dec 10 '18 at 13:44
Roloka
566
566
add a comment |
add a comment |
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