- #1

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Let A be a m by m matrix with characteristic polynomial

[tex]ch(x)=(x-\lambda)^{m}[/tex]

I want to show that there exist distinct vectors, v1 to vm, not zero, such that:

[tex]A\vec{v}_1=\lambda \vec{v}_1[/tex]

[tex]A\vec{v}_2=\lambda \vec{v}_2+\vec{v}_1[/tex]

[tex]A\vec{v}_3=\lambda \vec{v}_3+\vec{v}_2[/tex]

.

.

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[tex]A\vec{v}_m=\lambda \vec{v}_m+\vec{v}_{m-1}[/tex]

as long as these vectors exist, I can show that they are linearly independent, form a basis... bla bla bla and the Jordan form easily follows...But I just do not know how theses vectors must exist.