Let us consider the euclidean norm on $\mathbf{R}^2$. After some computations, I have obtained the following expression for the induced norm on 2 by 2 matrices.

$$ \left\lVert\pmatrix{a&b\cr c&d\cr}\right\rVert^2 = {1\over 2} \Bigl(\lvert a+ib\rvert^2+\lvert c+id\rvert^2+\lvert(a+ib)^2+(c+id)^2\rvert\Bigr). $$

This expression is new to me and I am wondering if there is a conceptual explanation for such a formula. Also, is there an analogous formula for higher dimensional matrices?

Let us consider the euclidean norm on $\mathbf{R}^2$. After some computations, I have obtained the following expression for the associated operator norm on 2 by 2 matrices.

$$ \left\lVert\pmatrix{a&b\cr c&d\cr}\right\rVert^2 = {1\over 2} \Bigl(\lvert a+ib\rvert^2+\lvert c+id\rvert^2+\lvert(a+ib)^2+(c+id)^2\rvert\Bigr). $$

This expression is new to me and I am wondering if there is a conceptual explanation for such a formula. Also, is there an analogous formula for higher dimensional matrices?

Let us consider the euclidean norm on ${\bf R}^2$. After some computations, I have obtained the following expression for the induced norm on 2 by 2 matrices.

$$ \left\|\pmatrix{a&b\cr c&d\cr}\right\|^2 = {1\over 2} \Bigl(|a+ib|^2+|c+id|^2+|(a+ib)^2+(c+id)^2|\Bigr) $$

This expression is new to me and I am wondering if there is a conceptual explanation for such a formula. Also, is there an analogous formula for higher dimensional matrices?

Let us consider the euclidean norm on $\mathbf{R}^2$. After some computations, I have obtained the following expression for the induced norm on 2 by 2 matrices.

$$ \left\lVert\pmatrix{a&b\cr c&d\cr}\right\rVert^2 = {1\over 2} \Bigl(\lvert a+ib\rvert^2+\lvert c+id\rvert^2+\lvert(a+ib)^2+(c+id)^2\rvert\Bigr). $$

This expression is new to me and I am wondering if there is a conceptual explanation for such a formula. Also, is there an analogous formula for higher dimensional matrices?