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Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?


Determinant of identity minus product of orthogonal matrix and rank-$1$ matrixDerivative for eigenvalue with respect to 1st / 2nd / 3rd invariant of a matrixFind the square of a 3x3 matrix that equals the negative Identity3 matrixWhy is the matrix product of 2 orthogonal matrices also an orthogonal matrix?Integral eigenvalues of a matrixIf $lambda$ is an eigenvalue of an orthogonal matrix $C$, prove $frac1lambda$ is an eigenvalue of $C^T$Commutating nilpotent operatorsShow that characteristic equation of an orthogonal matrix is a reciprocal equation.Characteristic polynomial roots of $ 3times 3$ orthogonal matrix.characteristic polynomial of skew Hermitian matrix coefficients are real??













1












$begingroup$


The characteristic equation of matrix A is
$$lambda ^3 - I_1lambda^2 + I_2lambda-I_3 = 0 $$



For orthogonal matrix
$$I_3 = det(A) = pm1$$
$$I_1 = tr(A)$$



Taking examples of orthogonal matrices, it looks like $I_1 = I_2$. Is this true always for an orthogonal matrix? Is there some proof?










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  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
    $endgroup$
    – dantopa
    Mar 12 at 16:51















1












$begingroup$


The characteristic equation of matrix A is
$$lambda ^3 - I_1lambda^2 + I_2lambda-I_3 = 0 $$



For orthogonal matrix
$$I_3 = det(A) = pm1$$
$$I_1 = tr(A)$$



Taking examples of orthogonal matrices, it looks like $I_1 = I_2$. Is this true always for an orthogonal matrix? Is there some proof?










share|cite|improve this question









New contributor




James is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$











  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
    $endgroup$
    – dantopa
    Mar 12 at 16:51













1












1








1





$begingroup$


The characteristic equation of matrix A is
$$lambda ^3 - I_1lambda^2 + I_2lambda-I_3 = 0 $$



For orthogonal matrix
$$I_3 = det(A) = pm1$$
$$I_1 = tr(A)$$



Taking examples of orthogonal matrices, it looks like $I_1 = I_2$. Is this true always for an orthogonal matrix? Is there some proof?










share|cite|improve this question









New contributor




James is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




The characteristic equation of matrix A is
$$lambda ^3 - I_1lambda^2 + I_2lambda-I_3 = 0 $$



For orthogonal matrix
$$I_3 = det(A) = pm1$$
$$I_1 = tr(A)$$



Taking examples of orthogonal matrices, it looks like $I_1 = I_2$. Is this true always for an orthogonal matrix? Is there some proof?







linear-algebra matrices trace orthogonal-matrices






share|cite|improve this question









New contributor




James is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




James is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited Mar 12 at 21:31









Yanior Weg

2,51911246




2,51911246






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asked Mar 12 at 16:11









JamesJames

62




62




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James is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






James is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
    $endgroup$
    – dantopa
    Mar 12 at 16:51
















  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
    $endgroup$
    – dantopa
    Mar 12 at 16:51















$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 12 at 16:51




$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 12 at 16:51










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