Norms (1,2,inf) of A=kI


Norms (1,2,inf) of A=kI

Postby Guest » Tue Nov 24, 2020 5:12 pm

Let the matrix A of size N x N, given by A = k I, be the identity matrix of size N x N and k a strictly positive real.

How would I calculate;

-the norms (1, 2, inf) of the matrix;

-its number of packaging in each of the standards 1, 2, inf

I'm truly lost. I do have the formulas of norms (1,2,inf) for a matrix, but I can figure it out.

Re: Norms (1,2,inf) of A=kI

Postby Guest » Mon Nov 30, 2020 2:13 pm

I am puzzled by this. You say you know the definitions. This matrix, a diagonal matrix with the same number, "k", on the diagonal, is a particularly trivial matrix!

You know that [tex]||A||_1= max \sum_i |a_{ij}|[/tex]. That sum is over each column but since each column has one k and n-1 0s each sum is just k and the maximum is k.

[tex]||A||_\infty[/tex], conversely, the maximum of the sums over the rows but, again, every row sum is k so the maximum is k.

[tex]||A||_2[tex] is the absolute value of the largest eigenvalue. The diagonal matrix with "k"s on the diagonal has all eigenvalues k so the largest eigenvalue is k. This norm is, again, k.

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