How can I create a change in row entries of a matrix in which characteristic polynomial be same for two matrix even by this change to get nice arrangment? (for example: the second element of last row of matrix 3*3 is zero but I want to change this value.)
characteristic polynomial of a matrix
0
$\begingroup$
matrices
-
0I mean, when expanding determinants it is usually to your advantage to have as many zeros as possible, is it not? To clarify the point about block forms: If, following to Gerry Myerson's answer, we conjugate your matrix $A$ with the matrix B=\pmatrix{1&0&0\cr0&0&1\cr0&1&0\cr} we get B^{-1}AB=\pmatrix{2&1&0\cr3&3&0\cr0&0&7\cr}. Isn't it easier to calculate the characteristic polynomial of this matrix instead of one, where one or several zeros are replaced by something else? – 2012-01-16
1 Answers
1
Matrices $A$ and $B^{-1}AB$ have the same characteristic polynomial, so given a matrix $A$ that you don't like because it has an annoying zero in it, you could just pick some random invertible matrix $B$ and then replace $A$ with $B^{-1}AB$. But I have to agree with Jyrki's comment; I don't see why having a zero in row 3, column 2 makes it hard to find the Jordan form.