# polar

polar decomposition of a square matrix

### Syntax

`[Rho, Theta] = polar(A)`

### Arguments

A

a real or complex square matrix.

Rho

a real matrix.

Theta

a real or complex matrix.

### Description

[Rho, Theta] = polar(A) returns the polar form of `A` i.e. `A = Rho * expm(%i*Theta)` with `Rho` symmetric ≥ 0 and `Theta` hermitian ≥ 0.

### Examples

```// Build an input matrix A:
Theta_0 = toeplitz(1.5:-0.5:0)
Rho_0 = toeplitz(1:4)
A = Rho_0 * expm(%i*Theta_0)

// Compute its polar matricial components:
[Rho, Theta] = polar(A);
clean(Rho)    // is real symetric
clean(Theta)  // is hermitian

// Check that the computed decomposition builds A as well
norm(A - Rho*expm(%i*Theta), 1)```
```--> Theta_0 = toeplitz(1.5:-0.5:0)
Theta_0  =
1.5   1.    0.5   0.
1.    1.5   1.    0.5
0.5   1.    1.5   1.
0.    0.5   1.    1.5

--> Rho_0 = toeplitz(1:4)
Rho_0  =
1.   2.   3.   4.
2.   1.   2.   3.
3.   2.   1.   2.
4.   3.   2.   1.

--> A = Rho_0 * expm(%i*Theta_0)
A  =
-1.2699509 - 2.1374364i  -2.6833779 - 1.9074687i  -3.0456968 - 0.7713555i  -1.5838266 + 0.7745137i
-0.3831402 - 0.8547475i  -2.7866925 - 1.5182796i  -2.3759306 - 0.8785558i  -0.745459  + 0.2813657i
-0.745459  + 0.2813657i  -2.3759306 - 0.8785558i  -2.7866925 - 1.5182796i  -0.3831402 - 0.8547475i
-1.5838266 + 0.7745137i  -3.0456968 - 0.7713555i  -2.6833779 - 1.9074687i  -1.2699509 - 2.1374364i

--> // Compute its polar matricial components:
--> [Rho, Theta] = polar(A);

--> clean(Rho)    // is real symetric
ans  =
4.441742  + 0.i   2.4611614 + 0.i   1.4611614 + 0.i   1.441742  + 0.i
2.4611614 + 0.i   2.6572775 + 0.i   1.6572775 + 0.i   1.4611614 + 0.i
1.4611614 + 0.i   1.6572775 + 0.i   2.6572775 + 0.i   2.4611614 + 0.i
1.441742  + 0.i   1.4611614 + 0.i   2.4611614 + 0.i   4.441742  + 0.i

--> clean(Theta)  // is hermitian
ans  =
-0.6115539 + 0.i          0.2232029 + 1.3108917i  -0.2767971 + 1.3108917i   1.0300387 + 0.i
0.2232029 - 1.3108917i  -1.1008351 + 0.i          1.5407576 + 0.i         -0.2767971 - 1.3108917i
-0.2767971 - 1.3108917i   1.5407576 + 0.i         -1.1008351 + 0.i          0.2232029 - 1.3108917i
1.0300387 + 0.i         -0.2767971 + 1.3108917i   0.2232029 + 1.3108917i  -0.6115539 + 0.i

--> // Check that the computed decomposition builds A as well
--> norm(A - Rho*expm(%i*Theta), 1)
ans  =
1.435D-14
```