This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise inverse hyperbolic cosine use ArrayBase::acosh .

Returns

an expression of the matrix inverse hyperbolic cosine of *this.

Returns

an expression of the adjoint (i.e. conjugate transpose) of *this.

Example:

Matrix2cf m = Matrix2cf::Random();
cout << "Here is the 2x2 complex matrix m:" << endl << m << endl;
cout << "Here is the adjoint of m:" << endl << m.adjoint() << endl;

Output:

Here is the 2x2 complex matrix m:
 (-0.211,0.68) (-0.605,0.823)
 (0.597,0.566)  (0.536,-0.33)
Here is the adjoint of m:
 (-0.211,-0.68)  (0.597,-0.566)
(-0.605,-0.823)    (0.536,0.33)

Warning

If you want to replace a matrix by its own adjoint, do NOT do this:

m = m.adjoint(); // bug!!! caused by aliasing effect

Instead, use the adjointInPlace() method:

m.adjointInPlace();

which gives Eigen good opportunities for optimization, or alternatively you can also do:

m = m.adjoint().eval();

See also

adjointInPlace(), transpose(), conjugate(), class Transpose, class internal::scalar_conjugate_op

This is the "in place" version of adjoint(): it replaces *this by its own transpose. Thus, doing

m.adjointInPlace();

has the same effect on m as doing

m = m.adjoint().eval();

and is faster and also safer because in the latter line of code, forgetting the eval() results in a bug caused by aliasing.

Notice however that this method is only useful if you want to replace a matrix by its own adjoint. If you just need the adjoint of a matrix, use adjoint().

Note

if the matrix is not square, then *this must be a resizable matrix. This excludes (non-square) fixed-size matrices, block-expressions and maps.

See also

transpose(), adjoint(), transposeInPlace()

Apply the elementary reflector H given by \( H = I - tau v v^*\) with \( v^T = [1 essential^T] \) from the left to a vector or matrix.

On input:

Parameters

essential	the essential part of the vector v
tau	the scaling factor of the Householder transformation
workspace	a pointer to working space with at least this->cols() entries

See also

MatrixBase::makeHouseholder(), MatrixBase::makeHouseholderInPlace(), MatrixBase::applyHouseholderOnTheRight()

Apply the elementary reflector H given by \( H = I - tau v v^*\) with \( v^T = [1 essential^T] \) from the right to a vector or matrix.

On input:

Parameters

essential	the essential part of the vector v
tau	the scaling factor of the Householder transformation
workspace	a pointer to working space with at least this->rows() entries

See also

MatrixBase::makeHouseholder(), MatrixBase::makeHouseholderInPlace(), MatrixBase::applyHouseholderOnTheLeft()

replaces *this by other * *this.

Example:

Matrix3f A = Matrix3f::Random(3,3), B;
B << 0,1,0,  
     0,0,1,  
     1,0,0;
cout << "At start, A = " << endl << A << endl;
A.applyOnTheLeft(B); 
cout << "After applyOnTheLeft, A = " << endl << A << endl;

Output:

At start, A = 
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
After applyOnTheLeft, A = 
-0.211  0.823  0.536
 0.566 -0.605 -0.444
  0.68  0.597  -0.33

This is defined in the Jacobi module.

#include <Eigen/Jacobi> 

Applies the rotation in the plane j to the rows p and q of *this, i.e., it computes B = J * B, with \( B = \left ( \begin{array}{cc} \text{*this.row}(p) \\ \text{*this.row}(q) \end{array} \right ) \).

See also

class JacobiRotation, MatrixBase::applyOnTheRight(), internal::apply_rotation_in_the_plane()

replaces *this by *this * other. It is equivalent to MatrixBase::operator*=().

Example:

Matrix3f A = Matrix3f::Random(3,3), B;
B << 0,1,0,  
     0,0,1,  
     1,0,0;
cout << "At start, A = " << endl << A << endl;
A *= B;
cout << "After A *= B, A = " << endl << A << endl;
A.applyOnTheRight(B);  // equivalent to A *= B
cout << "After applyOnTheRight, A = " << endl << A << endl;

Output:

At start, A = 
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
After A *= B, A = 
 -0.33   0.68  0.597
 0.536 -0.211  0.823
-0.444  0.566 -0.605
After applyOnTheRight, A = 
 0.597  -0.33   0.68
 0.823  0.536 -0.211
-0.605 -0.444  0.566

Returns

an Array expression of this matrix

See also

ArrayBase::matrix()

Returns

a const Array expression of this matrix

See also

ArrayBase::matrix()

Returns

a pseudo-expression of a diagonal matrix with *this as vector of diagonal coefficients

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

Example:

cout << Matrix3i(Vector3i(2,5,6).asDiagonal()) << endl;

Output:

2 0 0
0 5 0
0 0 6

See also

class DiagonalWrapper, class DiagonalMatrix, diagonal(), isDiagonal()

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise inverse hyperbolic sine use ArrayBase::asinh .

Returns

an expression of the matrix inverse hyperbolic sine of *this.

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise inverse hyperbolic cosine use ArrayBase::atanh .

Returns

an expression of the matrix inverse hyperbolic cosine of *this.

This is defined in the SVD module.

#include <Eigen/SVD> 

Returns

the singular value decomposition of *this computed by Divide & Conquer algorithm

See also

class BDCSVD

Returns

the l2 norm of *this using the Blue's algorithm. A Portable Fortran Program to Find the Euclidean Norm of a Vector, ACM TOMS, Vol 4, Issue 1, 1978.

For architecture/scalar types without vectorization, this version is much faster than stableNorm(). Otherwise the stableNorm() is faster.

See also

norm(), stableNorm(), hypotNorm()

Returns

the column-pivoting Householder QR decomposition of *this.

See also

class ColPivHouseholderQR

Returns

the complete orthogonal decomposition of *this.

See also

class CompleteOrthogonalDecomposition

This is defined in the LU module.

#include <Eigen/LU> 

Computation of matrix inverse and determinant, with invertibility check.

This is only for fixed-size square matrices of size up to 4x4.

Notice that it will trigger a copy of input matrix when trying to do the inverse in place.

Parameters

inverse	Reference to the matrix in which to store the inverse.
determinant	Reference to the variable in which to store the determinant.
invertible	Reference to the bool variable in which to store whether the matrix is invertible.
absDeterminantThreshold	Optional parameter controlling the invertibility check. The matrix will be declared invertible if the absolute value of its determinant is greater than this threshold.

Example:

Matrix3d m = Matrix3d::Random();
cout << "Here is the matrix m:" << endl << m << endl;
Matrix3d inverse;
bool invertible;
double determinant;
m.computeInverseAndDetWithCheck(inverse,determinant,invertible);
cout << "Its determinant is " << determinant << endl;
if(invertible) {
  cout << "It is invertible, and its inverse is:" << endl << inverse << endl;
}
else {
  cout << "It is not invertible." << endl;
}

Output:

Here is the matrix m:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Its determinant is 0.209
It is invertible, and its inverse is:
-0.199   2.23   2.84
  1.01 -0.555  -1.42
 -1.62   3.59   3.29

See also

inverse(), computeInverseWithCheck()

This is defined in the LU module.

#include <Eigen/LU> 

Computation of matrix inverse, with invertibility check.

This is only for fixed-size square matrices of size up to 4x4.

Notice that it will trigger a copy of input matrix when trying to do the inverse in place.

Parameters

inverse	Reference to the matrix in which to store the inverse.
invertible	Reference to the bool variable in which to store whether the matrix is invertible.
absDeterminantThreshold	Optional parameter controlling the invertibility check. The matrix will be declared invertible if the absolute value of its determinant is greater than this threshold.

Example:

Matrix3d m = Matrix3d::Random();
cout << "Here is the matrix m:" << endl << m << endl;
Matrix3d inverse;
bool invertible;
m.computeInverseWithCheck(inverse,invertible);
if(invertible) {
  cout << "It is invertible, and its inverse is:" << endl << inverse << endl;
}
else {
  cout << "It is not invertible." << endl;
}

Output:

Here is the matrix m:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
It is invertible, and its inverse is:
-0.199   2.23   2.84
  1.01 -0.555  -1.42
 -1.62   3.59   3.29

See also

inverse(), computeInverseAndDetWithCheck()

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise cosine use ArrayBase::cos .

Returns

an expression of the matrix cosine of *this.

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise hyperbolic cosine use ArrayBase::cosh .

Returns

an expression of the matrix hyperbolic cosine of *this.

This is defined in the LU module.

#include <Eigen/LU> 

Returns

the determinant of this matrix

Returns

an expression of the main diagonal of the matrix *this

*this is not required to be square.

Example:

Matrix3i m = Matrix3i::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here are the coefficients on the main diagonal of m:" << endl
     << m.diagonal() << endl;

Output:

Here is the matrix m:
 7  6 -3
-2  9  6
 6 -6 -5
Here are the coefficients on the main diagonal of m:
 7
 9
-5

See also

class Diagonal

Returns

an expression of the DiagIndex-th sub or super diagonal of the matrix *this

*this is not required to be square.

The template parameter DiagIndex represent a super diagonal if DiagIndex > 0 and a sub diagonal otherwise. DiagIndex == 0 is equivalent to the main diagonal.

Example:

Matrix4i m = Matrix4i::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here are the coefficients on the 1st super-diagonal and 2nd sub-diagonal of m:" << endl
     << m.diagonal<1>().transpose() << endl
     << m.diagonal<-2>().transpose() << endl;

Output:

Here is the matrix m:
 7  9 -5 -3
-2 -6  1  0
 6 -3  0  9
 6  6  3  9
Here are the coefficients on the 1st super-diagonal and 2nd sub-diagonal of m:
9 1 9
6 6

See also

MatrixBase::diagonal(), class Diagonal

This is the const version of diagonal().

This is the const version of diagonal<int>().

Returns

an expression of the DiagIndex-th sub or super diagonal of the matrix *this

*this is not required to be square.

The template parameter DiagIndex represent a super diagonal if DiagIndex > 0 and a sub diagonal otherwise. DiagIndex == 0 is equivalent to the main diagonal.

Example:

Matrix4i m = Matrix4i::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here are the coefficients on the 1st super-diagonal and 2nd sub-diagonal of m:" << endl
     << m.diagonal(1).transpose() << endl
     << m.diagonal(-2).transpose() << endl;

Output:

Here is the matrix m:
 7  9 -5 -3
-2 -6  1  0
 6 -3  0  9
 6  6  3  9
Here are the coefficients on the 1st super-diagonal and 2nd sub-diagonal of m:
9 1 9
6 6

See also

MatrixBase::diagonal(), class Diagonal

This is the const version of diagonal(Index).

Returns

the size of the main diagonal, which is min(rows(),cols()).

See also

rows(), cols(), SizeAtCompileTime.

Returns

the dot product of *this with other.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

Note

If the scalar type is complex numbers, then this function returns the hermitian (sesquilinear) dot product, conjugate-linear in the first variable and linear in the second variable.

See also

squaredNorm(), norm()

Computes the eigenvalues of a matrix.

Returns

Column vector containing the eigenvalues.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues> 

This function computes the eigenvalues with the help of the EigenSolver class (for real matrices) or the ComplexEigenSolver class (for complex matrices).

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix.

The SelfAdjointView class provides a better algorithm for selfadjoint matrices.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
VectorXcd eivals = ones.eigenvalues();
cout << "The eigenvalues of the 3x3 matrix of ones are:" << endl << eivals << endl;

Output:

The eigenvalues of the 3x3 matrix of ones are:
(-5.31e-17,0)
        (3,0)
        (0,0)

See also

EigenSolver::eigenvalues(), ComplexEigenSolver::eigenvalues(), SelfAdjointView::eigenvalues()

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise exponential use ArrayBase::exp .

Returns

an expression of the matrix exponential of *this.

Returns

an expression of *this with forced aligned access

See also

forceAlignedAccessIf(), class ForceAlignedAccess

Returns

an expression of *this with forced aligned access

See also

forceAlignedAccessIf(),class ForceAlignedAccess

Returns

an expression of *this with forced aligned access if Enable is true.

See also

forceAlignedAccess(), class ForceAlignedAccess

Returns

an expression of *this with forced aligned access if Enable is true.

See also

forceAlignedAccess(), class ForceAlignedAccess

Returns

the full-pivoting Householder QR decomposition of *this.

See also

class FullPivHouseholderQR

This is defined in the LU module.

#include <Eigen/LU> 

Returns

the full-pivoting LU decomposition of *this.

See also

class FullPivLU

Returns

the Householder QR decomposition of *this.

See also

class HouseholderQR

Returns

the l2 norm of *this avoiding undeflow and overflow. This version use a concatenation of hypot() calls, and it is very slow.

See also

norm(), stableNorm()

Returns

an expression of the identity matrix (not necessarily square).

This variant is only for fixed-size MatrixBase types. For dynamic-size types, you need to use the variant taking size arguments.

Example:

cout << Matrix<double, 3, 4>::Identity() << endl;

Output:

1 0 0 0
0 1 0 0
0 0 1 0

See also

Identity(Index,Index), setIdentity(), isIdentity()

Returns

an expression of the identity matrix (not necessarily square).

The parameters rows and cols are the number of rows and of columns of the returned matrix. Must be compatible with this MatrixBase type.

This variant is meant to be used for dynamic-size matrix types. For fixed-size types, it is redundant to pass rows and cols as arguments, so Identity() should be used instead.

Example:

cout << MatrixXd::Identity(4, 3) << endl;

Output:

1 0 0
0 1 0
0 0 1
0 0 0

See also

Identity(), setIdentity(), isIdentity()

This is defined in the LU module.

#include <Eigen/LU> 

Returns

the matrix inverse of this matrix.

For small fixed sizes up to 4x4, this method uses cofactors. In the general case, this method uses class PartialPivLU.

Note

This matrix must be invertible, otherwise the result is undefined. If you need an invertibility check, do the following:

• for fixed sizes up to 4x4, use computeInverseAndDetWithCheck().
• for the general case, use class FullPivLU.

Example:

Matrix3d m = Matrix3d::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Its inverse is:" << endl << m.inverse() << endl;

Output:

Here is the matrix m:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
Its inverse is:
-0.199   2.23   2.84
  1.01 -0.555  -1.42
 -1.62   3.59   3.29

See also

computeInverseAndDetWithCheck()

Returns

true if *this is approximately equal to a diagonal matrix, within the precision given by prec.

Example:

Matrix3d m = 10000 * Matrix3d::Identity();
m(0,2) = 1;
cout << "Here's the matrix m:" << endl << m << endl;
cout << "m.isDiagonal() returns: " << m.isDiagonal() << endl;
cout << "m.isDiagonal(1e-3) returns: " << m.isDiagonal(1e-3) << endl;
 

Output:

Here's the matrix m:
1e+04     0     1
    0 1e+04     0
    0     0 1e+04
m.isDiagonal() returns: 0
m.isDiagonal(1e-3) returns: 1

See also

asDiagonal()

Returns

true if *this is approximately equal to the identity matrix (not necessarily square), within the precision given by prec.

Example:

Matrix3d m = Matrix3d::Identity();
m(0,2) = 1e-4;
cout << "Here's the matrix m:" << endl << m << endl;
cout << "m.isIdentity() returns: " << m.isIdentity() << endl;
cout << "m.isIdentity(1e-3) returns: " << m.isIdentity(1e-3) << endl;

Output:

Here's the matrix m:
     1      0 0.0001
     0      1      0
     0      0      1
m.isIdentity() returns: 0
m.isIdentity(1e-3) returns: 1

See also

class CwiseNullaryOp, Identity(), Identity(Index,Index), setIdentity()

Returns

true if *this is approximately equal to a lower triangular matrix, within the precision given by prec.

See also

isUpperTriangular()

Returns

true if *this is approximately orthogonal to other, within the precision given by prec.

Example:

Vector3d v(1,0,0);
Vector3d w(1e-4,0,1);
cout << "Here's the vector v:" << endl << v << endl;
cout << "Here's the vector w:" << endl << w << endl;
cout << "v.isOrthogonal(w) returns: " << v.isOrthogonal(w) << endl;
cout << "v.isOrthogonal(w,1e-3) returns: " << v.isOrthogonal(w,1e-3) << endl;

Output:

Here's the vector v:
1
0
0
Here's the vector w:
0.0001
     0
     1
v.isOrthogonal(w) returns: 0
v.isOrthogonal(w,1e-3) returns: 1

Returns

true if *this is approximately an unitary matrix, within the precision given by prec. In the case where the Scalar type is real numbers, a unitary matrix is an orthogonal matrix, whence the name.

Note

This can be used to check whether a family of vectors forms an orthonormal basis. Indeed, m.isUnitary() returns true if and only if the columns (equivalently, the rows) of m form an orthonormal basis.

Example:

Matrix3d m = Matrix3d::Identity();
m(0,2) = 1e-4;
cout << "Here's the matrix m:" << endl << m << endl;
cout << "m.isUnitary() returns: " << m.isUnitary() << endl;
cout << "m.isUnitary(1e-3) returns: " << m.isUnitary(1e-3) << endl;

Output:

Here's the matrix m:
     1      0 0.0001
     0      1      0
     0      0      1
m.isUnitary() returns: 0
m.isUnitary(1e-3) returns: 1

Returns

true if *this is approximately equal to an upper triangular matrix, within the precision given by prec.

See also

isLowerTriangular()

This is defined in the SVD module.

#include <Eigen/SVD> 

Returns

the singular value decomposition of *this computed by two-sided Jacobi transformations.

See also

class JacobiSVD

Returns

an expression of the matrix product of *this and other without implicit evaluation.

The returned product will behave like any other expressions: the coefficients of the product will be computed once at a time as requested. This might be useful in some extremely rare cases when only a small and no coherent fraction of the result's coefficients have to be computed.

Warning

This version of the matrix product can be much much slower. So use it only if you know what you are doing and that you measured a true speed improvement.

See also

operator*(const MatrixBase&)

This is defined in the Cholesky module.

#include <Eigen/Cholesky> 

Returns

the Cholesky decomposition with full pivoting without square root of *this

See also

SelfAdjointView::ldlt()

This is defined in the Cholesky module.

#include <Eigen/Cholesky> 

Returns

the LLT decomposition of *this

See also

SelfAdjointView::llt()

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise logarithm use ArrayBase::log .

Returns

an expression of the matrix logarithm of *this.

Returns

the coefficient-wise \( \ell^p \) norm of *this, that is, returns the p-th root of the sum of the p-th powers of the absolute values of the coefficients of *this. If p is the special value Eigen::Infinity, this function returns the \( \ell^\infty \) norm, that is the maximum of the absolute values of the coefficients of *this.

In all cases, if *this is empty, then the value 0 is returned.

Note

For matrices, this function does not compute the operator-norm. That is, if *this is a matrix, then its coefficients are interpreted as a 1D vector. Nonetheless, you can easily compute the 1-norm and \(\infty\)-norm matrix operator norms using partial reductions .

See also

norm()

This is defined in the LU module.

#include <Eigen/LU> 

Synonym of partialPivLu().

Returns

the partial-pivoting LU decomposition of *this.

See also

class PartialPivLU

Computes the elementary reflector H such that: \( H *this = [ beta 0 ... 0]^T \) where the transformation H is: \( H = I - tau v v^*\) and the vector v is: \( v^T = [1 essential^T] \)

On output:

Parameters

essential	the essential part of the vector v
tau	the scaling factor of the Householder transformation
beta	the result of H * *this

See also

MatrixBase::makeHouseholderInPlace(), MatrixBase::applyHouseholderOnTheLeft(), MatrixBase::applyHouseholderOnTheRight()

Computes the elementary reflector H such that: \( H *this = [ beta 0 ... 0]^T \) where the transformation H is: \( H = I - tau v v^*\) and the vector v is: \( v^T = [1 essential^T] \)

The essential part of the vector v is stored in *this.

On output:

Parameters

tau	the scaling factor of the Householder transformation
beta	the result of H * *this

See also

MatrixBase::makeHouseholder(), MatrixBase::applyHouseholderOnTheLeft(), MatrixBase::applyHouseholderOnTheRight()

Returns

a pseudo expression of *this with an operator= assuming no aliasing between *this and the source expression.

More precisely, noalias() allows to bypass the EvalBeforeAssignBit flag. Currently, even though several expressions may alias, only product expressions have this flag. Therefore, noalias() is only useful when the source expression contains a matrix product.

Here are some examples where noalias is useful:

D.noalias()  = A * B;
D.noalias() += A.transpose() * B;
D.noalias() -= 2 * A * B.adjoint();

On the other hand the following example will lead to a wrong result:

A.noalias() = A * B;

because the result matrix A is also an operand of the matrix product. Therefore, there is no alternative than evaluating A * B in a temporary, that is the default behavior when you write:

A = A * B;

See also

class NoAlias

Returns

, for vectors, the l2 norm of *this, and for matrices the Frobenius norm. In both cases, it consists in the square root of the sum of the square of all the matrix entries. For vectors, this is also equals to the square root of the dot product of *this with itself.

See also

lpNorm(), dot(), squaredNorm()

Normalizes the vector, i.e. divides it by its own norm.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

Warning

If the input vector is too small (i.e., this->norm()==0), then *this is left unchanged.

See also

norm(), normalized()

Returns

an expression of the quotient of *this by its own norm.

Warning

If the input vector is too small (i.e., this->norm()==0), then this function returns a copy of the input.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

norm(), normalize()

Returns

true if at least one pair of coefficients of *this and other are not exactly equal to each other.

Warning

When using floating point scalar values you probably should rather use a fuzzy comparison such as isApprox()

See also

isApprox(), operator==

Returns

the diagonal matrix product of *this by the diagonal matrix diagonal.

Returns

the matrix product of *this and other.

Note

If instead of the matrix product you want the coefficient-wise product, see Cwise::operator*().

See also

lazyProduct(), operator*=(const MatrixBase&), Cwise::operator*()

replaces *this by *this * other.

Returns

a reference to *this

Example:

Matrix3f A = Matrix3f::Random(3,3), B;
B << 0,1,0,  
     0,0,1,  
     1,0,0;
cout << "At start, A = " << endl << A << endl;
A *= B;
cout << "After A *= B, A = " << endl << A << endl;
A.applyOnTheRight(B);  // equivalent to A *= B
cout << "After applyOnTheRight, A = " << endl << A << endl;

Output:

At start, A = 
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
After A *= B, A = 
 -0.33   0.68  0.597
 0.536 -0.211  0.823
-0.444  0.566 -0.605
After applyOnTheRight, A = 
 0.597  -0.33   0.68
 0.823  0.536 -0.211
-0.605 -0.444  0.566

replaces *this by *this + other.

Returns

a reference to *this

replaces *this by *this - other.

Returns

a reference to *this

Special case of the template operator=, in order to prevent the compiler from generating a default operator= (issue hit with g++ 4.1)

Returns

true if each coefficients of *this and other are all exactly equal.

Warning

When using floating point scalar values you probably should rather use a fuzzy comparison such as isApprox()

See also

isApprox(), operator!=

Computes the L2 operator norm.

Returns

Operator norm of the matrix.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues> 

This function computes the L2 operator norm of a matrix, which is also known as the spectral norm. The norm of a matrix \( A \) is defined to be

\[ \|A\|_2 = \max_x \frac{\|Ax\|_2}{\|x\|_2} \]

where the maximum is over all vectors and the norm on the right is the Euclidean vector norm. The norm equals the largest singular value, which is the square root of the largest eigenvalue of the positive semi-definite matrix \( A^*A \).

The current implementation uses the eigenvalues of \( A^*A \), as computed by SelfAdjointView::eigenvalues(), to compute the operator norm of a matrix. The SelfAdjointView class provides a better algorithm for selfadjoint matrices.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
cout << "The operator norm of the 3x3 matrix of ones is "
     << ones.operatorNorm() << endl;

Output:

The operator norm of the 3x3 matrix of ones is 3

See also

SelfAdjointView::eigenvalues(), SelfAdjointView::operatorNorm()

This is defined in the LU module.

#include <Eigen/LU> 

Returns

the partial-pivoting LU decomposition of *this.

See also

class PartialPivLU

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise power to p use ArrayBase::pow .

Returns

an expression of the matrix power to p of *this.

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise power to p use ArrayBase::pow .

Returns

an expression of the matrix power to p of *this.

Returns

an expression of a symmetric/self-adjoint view extracted from the upper or lower triangular part of the current matrix

The parameter UpLo can be either Upper or Lower

Example:

Matrix3i m = Matrix3i::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the symmetric matrix extracted from the upper part of m:" << endl
     << Matrix3i(m.selfadjointView<Upper>()) << endl;
cout << "Here is the symmetric matrix extracted from the lower part of m:" << endl
     << Matrix3i(m.selfadjointView<Lower>()) << endl;

Output:

Here is the matrix m:
 7  6 -3
-2  9  6
 6 -6 -5
Here is the symmetric matrix extracted from the upper part of m:
 7  6 -3
 6  9  6
-3  6 -5
Here is the symmetric matrix extracted from the lower part of m:
 7 -2  6
-2  9 -6
 6 -6 -5

See also

class SelfAdjointView

This is the const version of MatrixBase::selfadjointView()

Writes the identity expression (not necessarily square) into *this.

Example:

Matrix4i m = Matrix4i::Zero();
m.block<3,3>(1,0).setIdentity();
cout << m << endl;

Output:

0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0

See also

class CwiseNullaryOp, Identity(), Identity(Index,Index), isIdentity()

Resizes to the given size, and writes the identity expression (not necessarily square) into *this.

Parameters

rows	the new number of rows
cols	the new number of columns

Example:

MatrixXf m;
m.setIdentity(3, 3);
cout << m << endl;

Output:

1 0 0
0 1 0
0 0 1

See also

MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Identity()

Set the coefficients of *this to the i-th unit (basis) vector.

Parameters

i	index of the unique coefficient to be set to 1

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Unit(Index,Index)

Resizes to the given newSize, and writes the i-th unit (basis) vector into *this.

Parameters

newSize	the new size of the vector
i	index of the unique coefficient to be set to 1

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::setIdentity(), class CwiseNullaryOp, MatrixBase::Unit(Index,Index)

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise sine use ArrayBase::sin .

Returns

an expression of the matrix sine of *this.

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise hyperbolic sine use ArrayBase::sinh .

Returns

an expression of the matrix hyperbolic sine of *this.

This function requires the unsupported MatrixFunctions module. To compute the coefficient-wise square root use ArrayBase::sqrt .

Returns

an expression of the matrix square root of *this.

Returns

, for vectors, the squared l2 norm of *this, and for matrices the squared Frobenius norm. In both cases, it consists in the sum of the square of all the matrix entries. For vectors, this is also equals to the dot product of *this with itself.

See also

dot(), norm(), lpNorm()

Returns

the l2 norm of *this avoiding underflow and overflow. This version use a blockwise two passes algorithm: 1 - find the absolute largest coefficient s 2 - compute \( s \Vert \frac{*this}{s} \Vert \) in a standard way

For architecture/scalar types supporting vectorization, this version is faster than blueNorm(). Otherwise the blueNorm() is much faster.

See also

norm(), blueNorm(), hypotNorm()

Normalizes the vector while avoid underflow and overflow

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

This method is analogue to the normalize() method, but it reduces the risk of underflow and overflow when computing the norm.

Warning

If the input vector is too small (i.e., this->norm()==0), then *this is left unchanged.

See also

stableNorm(), stableNormalized(), normalize()

Returns

an expression of the quotient of *this by its own norm while avoiding underflow and overflow.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

This method is analogue to the normalized() method, but it reduces the risk of underflow and overflow when computing the norm.

Warning

If the input vector is too small (i.e., this->norm()==0), then this function returns a copy of the input.

See also

stableNorm(), stableNormalize(), normalized()

Returns

the trace of *this, i.e. the sum of the coefficients on the main diagonal.

*this can be any matrix, not necessarily square.

See also

diagonal(), sum()

Returns

an expression of a triangular view extracted from the current matrix

The parameter Mode can have the following values: Upper, StrictlyUpper, UnitUpper, Lower, StrictlyLower, UnitLower.

Example:

Matrix3i m = Matrix3i::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the upper-triangular matrix extracted from m:" << endl
     << Matrix3i(m.triangularView<Eigen::Upper>()) << endl;
cout << "Here is the strictly-upper-triangular matrix extracted from m:" << endl
     << Matrix3i(m.triangularView<Eigen::StrictlyUpper>()) << endl;
cout << "Here is the unit-lower-triangular matrix extracted from m:" << endl
     << Matrix3i(m.triangularView<Eigen::UnitLower>()) << endl;
// FIXME need to implement output for triangularViews (Bug 885)

Output:

Here is the matrix m:
 7  6 -3
-2  9  6
 6 -6 -5
Here is the upper-triangular matrix extracted from m:
 7  6 -3
 0  9  6
 0  0 -5
Here is the strictly-upper-triangular matrix extracted from m:
 0  6 -3
 0  0  6
 0  0  0
Here is the unit-lower-triangular matrix extracted from m:
 1  0  0
-2  1  0
 6 -6  1

See also

class TriangularView

This is the const version of MatrixBase::triangularView()

Returns

an expression of the i-th unit (basis) vector.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

This variant is for fixed-size vector only.

See also

MatrixBase::Unit(Index,Index), MatrixBase::UnitX(), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()

Returns

an expression of the i-th unit (basis) vector.

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::Unit(Index), MatrixBase::UnitX(), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()

Returns

an expression of the W axis unit vector (0,0,0,1)

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()

Returns

an expression of the X axis unit vector (1{,0}^*)

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()

Returns

an expression of the Y axis unit vector (0,1{,0}^*)

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()

Returns

an expression of the Z axis unit vector (0,0,1{,0}^*)

This is only for vectors (either row-vectors or column-vectors), i.e. matrices which are known at compile-time to have either one row or one column.

See also

MatrixBase::Unit(Index,Index), MatrixBase::Unit(Index), MatrixBase::UnitY(), MatrixBase::UnitZ(), MatrixBase::UnitW()