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```// Matrix properties checks.
use approx::RelativeEq;
use num::{One, Zero};

use crate::base::allocator::Allocator;
use crate::base::dimension::{Dim, DimMin};
use crate::base::storage::Storage;
use crate::base::{DefaultAllocator, Matrix, Scalar, SquareMatrix};

impl<N: Scalar, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
/// Indicates if this is an empty matrix.
#[inline]
pub fn is_empty(&self) -> bool {
let (nrows, ncols) = self.shape();
nrows == 0 || ncols == 0
}

/// Indicates if this is a square matrix.
#[inline]
pub fn is_square(&self) -> bool {
let (nrows, ncols) = self.shape();
nrows == ncols
}

// FIXME: RelativeEq prevents us from using those methods on integer matrices…
/// Indicated if this is the identity matrix within a relative error of `eps`.
///
/// If the matrix is diagonal, this checks that diagonal elements (i.e. at coordinates `(i, i)`
/// for i from `0` to `min(R, C)`) are equal one; and that all other elements are zero.
#[inline]
pub fn is_identity(&self, eps: N::Epsilon) -> bool
where
N: Zero + One + RelativeEq,
N::Epsilon: Copy,
{
let (nrows, ncols) = self.shape();
let d;

if nrows > ncols {
d = ncols;

for i in d..nrows {
for j in 0..ncols {
if !relative_eq!(self[(i, j)], N::zero(), epsilon = eps) {
return false;
}
}
}
} else {
// nrows <= ncols
d = nrows;

for i in 0..nrows {
for j in d..ncols {
if !relative_eq!(self[(i, j)], N::zero(), epsilon = eps) {
return false;
}
}
}
}

// Off-diagonal elements of the sub-square matrix.
for i in 1..d {
for j in 0..i {
// FIXME: use unsafe indexing.
if !relative_eq!(self[(i, j)], N::zero(), epsilon = eps)
|| !relative_eq!(self[(j, i)], N::zero(), epsilon = eps)
{
return false;
}
}
}

// Diagonal elements of the sub-square matrix.
for i in 0..d {
if !relative_eq!(self[(i, i)], N::one(), epsilon = eps) {
return false;
}
}

true
}
}

impl<N: ComplexField, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
/// Checks that `Mᵀ × M = Id`.
///
/// In this definition `Id` is approximately equal to the identity matrix with a relative error
/// equal to `eps`.
#[inline]
pub fn is_orthogonal(&self, eps: N::Epsilon) -> bool
where
N: Zero + One + ClosedAdd + ClosedMul + RelativeEq,
S: Storage<N, R, C>,
N::Epsilon: Copy,
DefaultAllocator: Allocator<N, R, C> + Allocator<N, C, C>,
{
}
}

impl<N: RealField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S>
where DefaultAllocator: Allocator<N, D, D>
{
/// Checks that this matrix is orthogonal and has a determinant equal to 1.
#[inline]
pub fn is_special_orthogonal(&self, eps: N) -> bool
where
D: DimMin<D, Output = D>,
DefaultAllocator: Allocator<(usize, usize), D>,
{
self.is_square() && self.is_orthogonal(eps) && self.determinant() > N::zero()
}

/// Returns `true` if this matrix is invertible.
#[inline]
pub fn is_invertible(&self) -> bool {
// FIXME: improve this?
self.clone_owned().try_inverse().is_some()
}
}
```