The matrix notation of signum?

Question

The following question on a notation might look trivial but I am really not sure how to deal with it.

If I have a variable $x$, I could write out:

$$x=|x|\;\text {sgn} (x)$$

a notation that helps me with an operator for the signs that could point to $-1$, $0$ or $+1$.

But then I have a matrix $\bf X$ with elements $x_{i,j}$ while the equation above holds for each element $x_{i,j}$, simply

$$x_{i,j}=|x_{i,j}|\;\text {sgn} (x_{i,j})$$

How does the matrix notation for the equation above look like, in terms of a matrix of $\bf X$ (and not individual elements)?

Answer 1

In multidimensioned normed space (e.g. matrices, vectors...) you can define a generalisation of the "sign" by saying $$sg(x):=\frac{x}{\|x\|},\quad x\ne 0.$$ Then you will obtain that in a any matrix norm $$X=\|X\|\cdot sg(X).$$

Note, however, there's no widely accepted notation for applying a function to the matrix element-wise, so you're free to introduce your own.

Answer 2

It is more complicated than it looks. If you have a square complex matrix $A$ you can decompose it into a product $$A=UP$$ where $U$ is unitary and $P$ is semidefinite positive. It is a generalization of the decomposition of a complex number $z$ as $$z=e^{i\theta} \lvert z \rvert.$$ See Polar decomposition on Wikipedia.

P.S.: Of course this has nothing to do with the sign decomposition of individual entries of the matrix. For that one I am afraid that a unified notation does not exist, so you will have to craft one by yourself.

Answer 3

The matrix sign function is defined in several equivalent ways. The one closest to what you're asking is

$$\mathop{\rm sign}(A) = A(A^2)^{-1/2}.$$

You can also use Jordan decomposition, integral definition,...

To find more about this, I advise reading Higham's "Functions of Matrices: Theory and Computation" (mostly Chapter 5, but some of the other chapters also draw a connection to the sign function).

Answer 4

Define the element-wise abs() function of the matrix $X$ as follows $$\eqalign{ A &= {\rm abs}(X) \;\doteq\; {\rm sqrt}\big(X\odot X^*\big) \\ }$$ Also define an element-wise pseudoinverse $A^\oplus$ operation $$ A_{ij}^\oplus = \begin{cases} \;0\quad&{\rm if}\;A_{ij} = 0 \\ A_{ij}^{-1}\quad&{\rm otherwise} \\ \end{cases} $$ and use this to define an element-wise sign() function $$\eqalign{ S &= {\rm sign}(X) \;\doteq\; X \odot A^\oplus \\ }$$ where $\odot$ denotes the elementwise/Hadamard product and $X^*$ the complex conjugate.

With this definition, one has the following useful relationships $$\eqalign{ A\odot S &= X \\ A\odot A &= X\odot X^* \\ }$$

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All of which describes the element-wise invocation of the Julia sign function

S = sign.(X)