Is $Av_1,Av_2,Av_3$ orthogonal if you have eigenvector of $A^TA$

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Let $Ain M_3(mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal?



I only know that we need to prove that $(Av_1,Av_2)=0$ and $(Av_1,Av_3)=0$ but I write that $(Av_1,Av_2)=(Av_1)^TAv_2=v_1^TA^TAv_2=2v_1^Tv_2=2(v_1,v_2)=0$ but I do not know is this prove ok?






linear-algebra orthogonality orthonormal orthogonal-matrices







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    Yes, your proof is fine.
    – Kavi Rama Murthy
    Sep 4 at 7:55














up vote
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down vote

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Let $Ain M_3(mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal?



I only know that we need to prove that $(Av_1,Av_2)=0$ and $(Av_1,Av_3)=0$ but I write that $(Av_1,Av_2)=(Av_1)^TAv_2=v_1^TA^TAv_2=2v_1^Tv_2=2(v_1,v_2)=0$ but I do not know is this prove ok?






linear-algebra orthogonality orthonormal orthogonal-matrices







share|cite|improve this question

















  • 2




    Yes, your proof is fine.
    – Kavi Rama Murthy
    Sep 4 at 7:55












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





Let $Ain M_3(mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal?



I only know that we need to prove that $(Av_1,Av_2)=0$ and $(Av_1,Av_3)=0$ but I write that $(Av_1,Av_2)=(Av_1)^TAv_2=v_1^TA^TAv_2=2v_1^Tv_2=2(v_1,v_2)=0$ but I do not know is this prove ok?






linear-algebra orthogonality orthonormal orthogonal-matrices







share|cite|improve this question













Let $Ain M_3(mathbb R)$ and if $v_1,v_2,v_3$ orthonormed eigenvectors of matrix $A^TA$ and which eigenvalues is $1,2,3$ then vectors $Av_1,Av_2,Av_3$ is orthogonal?



I only know that we need to prove that $(Av_1,Av_2)=0$ and $(Av_1,Av_3)=0$ but I write that $(Av_1,Av_2)=(Av_1)^TAv_2=v_1^TA^TAv_2=2v_1^Tv_2=2(v_1,v_2)=0$ but I do not know is this prove ok?






linear-algebra orthogonality orthonormal orthogonal-matrices




linear-algebra orthogonality orthonormal orthogonal-matrices






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 4 at 7:50









Marko Škorić

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  • 2




    Yes, your proof is fine.
    – Kavi Rama Murthy
    Sep 4 at 7:55












  • 2




    Yes, your proof is fine.
    – Kavi Rama Murthy
    Sep 4 at 7:55







2




2




Yes, your proof is fine.
– Kavi Rama Murthy
Sep 4 at 7:55




Yes, your proof is fine.
– Kavi Rama Murthy
Sep 4 at 7:55















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