[Math] Matrix bracket notation

matricesnotation

I am reading a section in a book that talks about normal matrices, and I see the following:

A normal matrix is a matrix that commutes with its adjoint,

huh?

Eh? What do the brackets and the comma inside mean?

Best Answer

The notation means that $AA^{T}-A^{T}A$=0.

Related Solutions

Understand that a normal matrix commutes with its adjoint, but a diagonalizable matrix may not

Every diagonalizable matrix $N$ on a finite-dimensional vector space $X$ is normal with respect to some inner product on $X$. Indeed, if $B=\{ b_1,b_2,\cdots,b_n \}$ is a basis for $X$ such that $Nb_k=\lambda_k b_k$, then you can define an inner product $\langle \cdot,\cdot\rangle_{B}$ by

$$ \langle \alpha_1b_1+\cdots+\alpha_n b_n,\beta_1 b_1+\cdots+\beta_n b_n\rangle_{B}=\sum_{j=1}^{n} \alpha_j\beta_j^*, $$ and you have the adjoint $N^*$ of $N$ with respect to this inner product given by $$ N^*(\alpha_1 b_1+\cdots+\alpha_n b_n)=\alpha_1^*b_1+\cdots+\alpha_n^*b_n. $$ So $N^*N=NN^*$ because $N^*,N$ share eigenvectors. Furthermore, the basis of eigenvectors is orthonormal with respect to $\langle\cdot,\cdot\rangle_B$. So, normality is not more or less general than diagonalizability; it's a matter of choosing the right inner product in order to make a diagonalizable $N$ normal.

Are there rules for omitted bracket use

Are there strict rules for omitting brackets?

There aren't any strict rules, but the aim should be that of clarity and unambiguity. Unfortunately, what can be notationally crystal clear for one can also be totally unclear for someone else.

For short fractions in arguments (like the case of the $\sin$ function in (14)) it's common to omit brackets, and so don't take that as hint that the argument of the logarithm is the full expression.

However, if the difference is important for you—in the sense that you need to use those equations—I really suggest you to re-derive them, because in papers, in reporting long expressions, it's way too easy to make mistakes and I wouldn't blindly trust those equations.

Best Answer

Related Solutions

Understand that a normal matrix commutes with its adjoint, but a diagonalizable matrix may not

Are there rules for omitted bracket use

Related Question