Is the matrix norm of a $1$x$m$ matrix A the same as the vector norm of a $m$-dimensional row vector?

Question

Consider a matrix $A$ of order $1$x$m$ and a $m$-dimensional row vector $x=(x_1,x_2,\dots,x_m)$.

$||A||_{\infty}=$ maximum row sum. And $||x||_{\infty} =$ max$\{|x_1|,\dots,|x_n|\}$.

E.g. Let $A=[ 1 -1 ]$. Then $||A||_{\infty} = |1|+|-1|=2.$ But can I not think of $A$ as a $2$-dimensional row vector? In which case the vector norm would be max {|1|,|-1|}=1.

Answer 1

The vector and matrix norms coincide when the matrix is a column, not a row. (In matrix/vector operations, vectors are seen as columns.)

https://en.wikipedia.org/wiki/Matrix_norm#Matrix_norms_induced_by_vector_norms