#self doubt

Question

I found this statement in math blog.....can anyone please help on this…

For a upper triangular Matrix I tried to derive with Eigen values as 1,2,0,0,0 for 5*5 but I am not getting rank as 2 for this Matrix.

Answer 1

Dknights as We know Rank is Equal to  Non Zero eigenvalues. So if Here They are saying $n-2$ eigenvalues are Zero So non zero eigen Values are $2$ which is Equal To rank.

Proof for Non zero eigen Values is Rank – https://math.berkeley.edu/~hutching/teach/54-2017/svd-notes.pdf 

Comments

But this is not always true 

https://math.stackexchange.com/questions/3710069/is-the-rank-of-a-matrix-equal-to-the-number-of-non-zero-eigenvalues

 

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https://sites.math.washington.edu/~shinms/SP14-54/midterm-2-review.pdf  Dknights AS U SAID ITS UPPER TRIANGULAR SO IT WILL AUTOMATICALLY DIAGONALIZABLE, WHICH MAKES THE UPPER STATEMENT TRUE.

 

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https://math.stackexchange.com/questions/1558591/if-a-matrix-is-triangular-is-there-a-quicker-way-to-tell-if-it-is-can-be-diagon

But the rank is not coming 2 for 5*5 which example i took.
[1,4,5,6,7

0,2,3,4,5

0,0,0,6,5

0,0,0,0,7

0,0,0,0,0

]