I have been recently introduced to the idea of Jordan form of a matrix, and I have attempted to prove it. However, there is one step that I cannot prove. If I can prove the step, then everything else is done.(adsbygoogle = window.adsbygoogle || []).push({});

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]

.

.

.

[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.

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# Proof of existence of a Jordan form?

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