Matrices & Linear Algebra
Matrix Arithmetic
do basic arithmetic on matrices
{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}
{{2,-1,1},{0,-2,1},{1,-2,0}}.{x,y,z}
Matrix Operations
compute properties of a matrix
{{6, -7, 10}, {0, 3, -1}, {0, 5, -7}}
compute the rank of a matrix
rank {{6, -11, 13}, {4, -1, 3}, {3, 4, -2}}
compute the inverse of a matrix
inv {{10, -9, -12}, {7, -12, 11}, {-10, 10, 3}}
inverse {{a, b}, {c, d}}
{{2,3},{4,5}}^(-1)
compute the adjugate of a matrix
adjugate {{8,7,7},{6,9,2},{-6,9,-2}}
Trace
compute the trace of a matrix
tr {{9, -6, 7}, {-9, 4, 0}, {-8, -6, 4}}
tr {{a, b}, {c, d}}
Determinant
compute the determinant of a matrix
determinant of {{3,4},{2,1}}
det({{9, 3, 5}, {-6, -9, 7}, {-1, -8, 1}})
det {{a, b, c}, {d, e, f}, {g, h, j}}
Row Reduction
row reduce a matrix
row reduce {{2,1,0,-3},{3,-1,0,1},{1,4,-2,-5}}
row reduction calculator
Eigenvalues & Eigenvectors
compute the eigenvalues of a matrix
eigenvalues {{4,1},{2,-1}}
compute the eigenvectors of a matrix
eigenvectors {{1,0,0},{0,0,1},{0,1,0}}
compute the characteristic polynomial of a matrix
characteristic polynomial {{4,1},{2,-1}}
Diagonalization
diagonalize a matrix
diagonalize {{1,2},{3,4}}
compute the LU decomposition of a square matrix
LU decomposition of {{7,3,-11},{-6,7,10},{-11,2,-2}}
compute a singular value decomposition
SVD {{1,0,-1},{-2,1,4}}
compute a 2 x 2 rotation matrix
rotate 30 degrees
compute a 3 x 3 reflection matrix
reflect across x+y+z=1
Types of Matrices
determine whether a matrix has a specified property
Is {{3,-3},{-3,5}} positive definite?
get information about a type of matrix
Hilbert matrices
Hankel matrices
specify a size
5x5 Hilbert matrix
Vector Spaces
compute the row space of a matrix
row space {{1,2,-5},{-1,0,-1},{2,1,-1}}
compute the column space of a matrix
{{1,0,-1},{2,-1,3}} column space
compute the null space of a matrix
null space of {{1,0,-2,1},{2,-1,1,0},{0,2,-3,1}}
Linear Independence
determine whether a set of vectors is linearly independent
Are (2,-1) and (4,2) linearly independent?
linear independence (1,3,-2), (2,1,-3), (-3,6,3)
specify complex vectors
are (1,i), (i,-1) linearly independent?
specify vectors with one or more symbolic components