Less than or equal set to a given matrix

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Suppose that $M$ is a given positive definite $ntimes n$ real matrix. I'm interested in the following sets and their properties
beginalign*
S_1&equivNtext positive semidefinite, dimension ntimes n: M-Ntext is positive semidefinitesubset mathbbM(n),\
S_2&equivxtext dimension ntimes1: M-xx'text is positive semidefinitesubsetmathbbR^n.
endalign*
The problem collapses to triviality when $n=1$ but harder for me to grasp when $ngeq 2$. I will appreciate references to books and articles, with a preference for the former.




linear-algebra reference-request soft-question




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    The book “Problems and Theorems of Linear Algebra” by Prasolov may be relevant
    – Selflearner
    Jul 27 at 17:59










  • @Selflearner Thank you. I will check it out tonight.
    – yurnero
    Jul 27 at 22:59














up vote
1
down vote

favorite
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Suppose that $M$ is a given positive definite $ntimes n$ real matrix. I'm interested in the following sets and their properties
beginalign*
S_1&equivNtext positive semidefinite, dimension ntimes n: M-Ntext is positive semidefinitesubset mathbbM(n),\
S_2&equivxtext dimension ntimes1: M-xx'text is positive semidefinitesubsetmathbbR^n.
endalign*
The problem collapses to triviality when $n=1$ but harder for me to grasp when $ngeq 2$. I will appreciate references to books and articles, with a preference for the former.




linear-algebra reference-request soft-question




share|cite|improve this question















  • 1




    The book “Problems and Theorems of Linear Algebra” by Prasolov may be relevant
    – Selflearner
    Jul 27 at 17:59










  • @Selflearner Thank you. I will check it out tonight.
    – yurnero
    Jul 27 at 22:59












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Suppose that $M$ is a given positive definite $ntimes n$ real matrix. I'm interested in the following sets and their properties
beginalign*
S_1&equivNtext positive semidefinite, dimension ntimes n: M-Ntext is positive semidefinitesubset mathbbM(n),\
S_2&equivxtext dimension ntimes1: M-xx'text is positive semidefinitesubsetmathbbR^n.
endalign*
The problem collapses to triviality when $n=1$ but harder for me to grasp when $ngeq 2$. I will appreciate references to books and articles, with a preference for the former.




linear-algebra reference-request soft-question




share|cite|improve this question











Suppose that $M$ is a given positive definite $ntimes n$ real matrix. I'm interested in the following sets and their properties
beginalign*
S_1&equivNtext positive semidefinite, dimension ntimes n: M-Ntext is positive semidefinitesubset mathbbM(n),\
S_2&equivxtext dimension ntimes1: M-xx'text is positive semidefinitesubsetmathbbR^n.
endalign*
The problem collapses to triviality when $n=1$ but harder for me to grasp when $ngeq 2$. I will appreciate references to books and articles, with a preference for the former.





linear-algebra reference-request soft-question





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 27 at 17:30









yurnero

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  • 1




    The book “Problems and Theorems of Linear Algebra” by Prasolov may be relevant
    – Selflearner
    Jul 27 at 17:59










  • @Selflearner Thank you. I will check it out tonight.
    – yurnero
    Jul 27 at 22:59












  • 1




    The book “Problems and Theorems of Linear Algebra” by Prasolov may be relevant
    – Selflearner
    Jul 27 at 17:59










  • @Selflearner Thank you. I will check it out tonight.
    – yurnero
    Jul 27 at 22:59







1




1




The book “Problems and Theorems of Linear Algebra” by Prasolov may be relevant
– Selflearner
Jul 27 at 17:59




The book “Problems and Theorems of Linear Algebra” by Prasolov may be relevant
– Selflearner
Jul 27 at 17:59












@Selflearner Thank you. I will check it out tonight.
– yurnero
Jul 27 at 22:59




@Selflearner Thank you. I will check it out tonight.
– yurnero
Jul 27 at 22:59















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