## 1. Settings of Eigen

### 1.1 Here Eigen3

• Type sudo apt-get install libeigen3-dev to install Eigen3.
• It will be installed in the default directory /usr/include/eigen3.
• Just copy Eigen under /usr/include/eigen3 to the directory /usr/include.

 1 2  #include using namespace Eigen; 

### 1.3 Tips

MatrixXd m(r, c); “X” means “dynamic”, “d” means basic data type “double”， “r” and “c” indicate no. of rows an columns.

MatrixXf m(r, c); “X” also means “dynamic”，“f” means “float”.

MatrixXcd c(r, c); “X” means “dynamic”，“c” means “complex”, “d” means “double”.

## 2. Basic numerical operations

### 2.1 Definitions of matrix in Eigen

  1 2 3 4 5 6 7 8 9 10 11 12 13 14  Matrix M1; // Fix rows and columns, same as Matrix5d. Matrix M2; // Fix rows. Matrix C; // Same as MatrixXd. Matrix E; // Storing according to rows, default is column majority. Matrix3f P, Q, R; // Float maxtrix with size of 3 x 3. Vector3f x, y, z; // Column vetor with 3 x 1. RowVector3f a, b, c; // Row vector with 1 x 3. VectorXd v; // Dynamic double vectors. // In Eigen3 // In matlab v.size() // length(v) // Length of vector v. C.rows() // size(C,1) // Rows of matrix. C.cols() // size(C,2) // Columns of matrix. v(i) // v(i+1) // Indexing of vector. C(i,j) // C(i+1,j+1) // Indexing of matrix. 

### 2.2 Resize (reshape in matlab or python) and fill

 1 2  MatrixXd A(4, 4) = MatrixXdf:Random(4, 4); // Giving values. A.resize(8, 2); // Change the dimensions. 
 1 2 3 4 5 6 7  MaxtrixXd A(3, 3); A << 1, 2, 3, // Initialize A. The elements can also be 4, 5, 6, // matrices, which are stacked along cols 7, 8, 9; // and then the rows are stacked. MaxtrixXd B(3, 9); B << A, A, A; // B is three horizontally stacked A's. A.fill(11); // Fill A with all elements of 1. 

### 2.3 Special matrices in Eigen3

  1 2 3 4 5 6 7 8 9 10 11  // In Eigen3 // In matlab MatrixXd::Identity(rows,cols) // eye(rows, cols) C.setIdentity(rows,cols) // C = eye(rows,cols) MatrixXd::Zero(rows,cols) // zeros(rows,cols) C.setZero(rows,cols) // C = zeros(rows,cols) MatrixXd::Ones(rows,cols) // ones(rows,cols) C.setOnes(rows,cols) // C = ones(rows,cols) MatrixXd::Random(rows,cols) // rand(rows,cols)*2-1 C.setRandom(rows,cols) // C = rand(rows,cols)*2-1 VectorXd::LinSpaced(size,low,high) // linspace(low,high,size)' v.setLinSpaced(size,low,high) // v = linspace(low,high,size)' 

### 2.4 Blocks of matrix

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  // In Eigen3 // In matlab x.head(n) // x(1:n) x.head() // x(1:n) x.tail(n) // x(end - n + 1: end) x.tail() // x(end - n + 1: end) x.segment(i, n) // x(i+1 : i+n) x.segment(i) // x(i+1 : i+n) P.block(i, j, rows, cols) // P(i+1 : i+rows, j+1 : j+cols) P.block(i, j) // P(i+1 : i+rows, j+1 : j+cols) P.row(i) // P(i+1, :) P.col(j) // P(:, j+1) P.leftCols() // P(:, 1:cols) P.leftCols(cols) // P(:, 1:cols) P.middleCols(j) // P(:, j+1:j+cols) P.middleCols(j, cols) // P(:, j+1:j+cols) P.rightCols() // P(:, end-cols+1:end) P.rightCols(cols) // P(:, end-cols+1:end) P.topRows() // P(1:rows, :) P.topRows(rows) // P(1:rows, :) P.middleRows(i) // P(i+1:i+rows, :) P.middleRows(i, rows) // P(i+1:i+rows, :) P.bottomRows() // P(end-rows+1:end, :) P.bottomRows(rows) // P(end-rows+1:end, :) P.topLeftCorner(rows, cols) // P(1:rows, 1:cols) P.topRightCorner(rows, cols) // P(1:rows, end-cols+1:end) P.bottomLeftCorner(rows, cols) // P(end-rows+1:end, 1:cols) P.bottomRightCorner(rows, cols) // P(end-rows+1:end, end-cols+1:end) P.topLeftCorner() // P(1:rows, 1:cols) P.topRightCorner() // P(1:rows, end-cols+1:end) P.bottomLeftCorner() // P(end-rows+1:end, 1:cols) P.bottomRightCorner() // P(end-rows+1:end, end-cols+1:end) 

### 2.5 Swaping elements

 1 2 3  // In Eigen3 // In matlab R.row(i) = P.col(j); // R(i, :) = P(:, i) R.col(j1).swap(mat1.col(j2)); // R(:, [j1 j2]) = R(:, [j2, j1]) 

### 2.6 Tranpose

 1 2 3 4 5 6 7 8  // Views, transpose, etc; all read-write except for .adjoint(). // In Eigen3 // In matlab R.adjoint() // R' R.transpose() // R.' or conj(R') R.diagonal() // diag(R) x.asDiagonal() // diag(x) R.transpose().colwise().reverse(); // rot90(R) R.conjugate() // conj(R) 

### 2.7 Products of Matrices

 1 2 3 4 5 6 7  // Matrix-vector. Matrix-matrix. Matrix-scalar. y = M*x; R = P*Q; R = P*s; a = b*M; R = P - Q; R = s*P; a *= M; R = P + Q; R = P/s; R *= Q; R = s*P; R += Q; R *= s; R -= Q; R /= s; 

### 2.8 Element operations

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32  // Vectorized operations on each element independently // In Eigen3 // In matlab R = P.cwiseProduct(Q); // R = P .* Q R = P.array() * s.array();// R = P .* s R = P.cwiseQuotient(Q); // R = P ./ Q R = P.array() / Q.array();// R = P ./ Q R = P.array() + s.array();// R = P + s R = P.array() - s.array();// R = P - s R.array() += s; // R = R + s R.array() -= s; // R = R - s R.array() < Q.array(); // R < Q R.array() <= Q.array(); // R <= Q R.cwiseInverse(); // 1 ./ P R.array().inverse(); // 1 ./ P R.array().sin() // sin(P) R.array().cos() // cos(P) R.array().pow(s) // P .^ s R.array().square() // P .^ 2 R.array().cube() // P .^ 3 R.cwiseSqrt() // sqrt(P) R.array().sqrt() // sqrt(P) R.array().exp() // exp(P) R.array().log() // log(P) R.cwiseMax(P) // max(R, P) R.array().max(P.array()) // max(R, P) R.cwiseMin(P) // min(R, P) R.array().min(P.array()) // min(R, P) R.cwiseAbs() // abs(P) R.array().abs() // abs(P) R.cwiseAbs2() // abs(P.^2) R.array().abs2() // abs(P.^2) (R.array() < s).select(P,Q); // (R < s ? P : Q) 

### 2.9 Simplifying matrices

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  // Reductions. int r, c; // Eigen // Matlab R.minCoeff() // min(R(:)) R.maxCoeff() // max(R(:)) s = R.minCoeff(&r, &c) // [s, i] = min(R(:)); [r, c] = ind2sub(size(R), i); s = R.maxCoeff(&r, &c) // [s, i] = max(R(:)); [r, c] = ind2sub(size(R), i); R.sum() // sum(R(:)) R.colwise().sum() // sum(R) R.rowwise().sum() // sum(R, 2) or sum(R')' R.prod() // prod(R(:)) R.colwise().prod() // prod(R) R.rowwise().prod() // prod(R, 2) or prod(R')' R.trace() // trace(R) R.all() // all(R(:)) R.colwise().all() // all(R) R.rowwise().all() // all(R, 2) R.any() // any(R(:)) R.colwise().any() // any(R) R.rowwise().any() // any(R, 2) 

### 2.10 Dot product

 1 2 3 4 5 6  // Dot products, norms, etc. // Eigen // Matlab x.norm() // norm(x). x.squaredNorm() // dot(x, x) x.dot(y) // dot(x, y) x.cross(y) // cross(x, y) Requires #include 

### 2.11 Type convertion

 1 2 3 4 5 6 7 8   Type conversion // Eigen // Matlab A.cast(); // double(A) A.cast(); // single(A) A.cast(); // int32(A) A.real(); // real(A) A.imag(); // imag(A) // if the original type equals destination type, no work is done 

### 2.12 Linear equation system: $\bf{Ax} = \bf{b}$

  1 2 3 4 5 6 7 8 9 10 11  // Solve Ax = b. Result stored in x. Matlab: x = A \ b. x = A.ldlt().solve(b)); // #include LDLT: Inprovement of Cholesky x = A.llt() .solve(b)); // A sym. p.d. #include x = A.lu() .solve(b)); // Stable and fast. #include x = A.qr() .solve(b)); // No pivoting. #include x = A.svd() .solve(b)); // Stable, slowest. #include // .ldlt() -> .matrixL() and .matrixD() // .llt() -> .matrixL() // .lu() -> .matrixL() and .matrixU() // .qr() -> .matrixQ() and .matrixR() // .svd() -> .matrixU(), .singularValues(), and .matrixV() 

### 2.13 Eigensystem problems of real matrix

 1 2 3 4 5  // In Eigen3 // In matlab A.eigenvalues(); // eig(A); // Finding eigenvalues. EigenSolver eig(A); // [vec val] = eig(A); // Eigensolver. eig.eigenvalues(); // val // Eigenvalues. eig.eigenvectors(); // vec // Eigenvectors. 

## 3. Operations of complex matrix.

### 3.1 Definitions.

 1 2 3  MatrixXcd c(r, c); // Declaring a r x c complex matrix. c.real() << MatrixXd::Random(r, c); // Giving values to the real part. c.imag() << MatrixXd::Random(r, c); // Giving values to the imaginary part. 

### 3.2 Getting elements, real and imaginary parts.

 1 2 3  MatrixXcd array = c.array(); MatrixXd real = c.real(); MatrixXd imag = c.imag(); 

### 3.3 Eigensystem problems

 1 2 3 4 5 6  MatrixXcd M(r, c); // Declaration of a dynamic complex matrix of r x c. M.real() << MatrixXd::Random(r, c); // Giving a random matrix to the real part of matrix M. M.imag() << MatrixXd::Random(r, c); // Giving a random matrix to the imaginary part of matrix M. ComplexEigenSolver es(M); // Complex eigensolver. MatrixXcd val = es.eigenvalues(); // Finding eigenvalues. MatrixXcd vec = es.eigenvectors(); // Finding eigen vectors. 

### Bibliography

https://blog.csdn.net/Zzhouzhou237/article/details/105078647