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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proof of existence of a Jordan form?

**Physics Forums | Science Articles, Homework Help, Discussion**