Linear Algebra

det(m)

Computes the determinant of the matrix \(m\)

>>> det({{1, 2, 4}, {4, 0, 6}, {7, 8, 9}})

= 92

lndet(m)

Computes the logarithm of the absolute value of the determinant of the matrix \(m\). That is, \(\ln|\det(m)|\)

>>> lndet({{1, 2, 4}, {4, 0, 6}, {7, 8, 9}})

= 4.52178857704904

LU(m)

Computes the \(LU\) decomposition of the matrix \(m\).

>>> lndet({{1, 2, 4}, {4, 0, 6}, {7, 8, 9}})

= (LU = {{7, 8, 9}, {4/7, -32/7, 6/7}, {1/7, -3/16, 23/8}}, P = {2, 1, 0}, sign = -1)

LUsolve(A, b)

Solves the square system \(Ax = b\). Aliases: solve

>>> LUsolve({{1, 2, 4}, {4, 0, 6}, {7, 8, 9}}, {1, 2, 3})
= {{4/23}, {-1/46}, {5/23}}

>>> solve({{1, 6, 4}, {4, 0, 6}, {7, 8, 2}}, {4, 2, 3})
= {{-11/71}, {57/142}, {31/71}}

Cholesky(m)

Factorize the symmetric, positive-definite square matrix \(m\) into the Cholesky decomposition \(m = L L^T\)

>>> Cholesky{{9, 3, 0}, {3, 5, 2}, {0, 2, 17}}
= {{3, 3, 0}, {1, 2, 2}, {0, 1, 4}
>>> chol{{9, 3, 0}, {3, 5, 2}, {0, 2, 17}}
= {{3, 3, 0}, {1, 2, 2}, {0, 1, 4}

SVD(m)

Factorizes the \(M \times N\) matrix \(m\) into the singular value decomposition \(m = U S V^T\) for \(M \ge N\).

>>> SVD{{4, 0}, {0, -1}}
= (U = {{1, 0}, {0, 1}}, S = {{4}, {1}}, V = {{1, 0}, {0, -1}})
>>> SVD{{3, 0}, {0, 2}}
= (U = {{1, 0}, {0, 1}}, S = {{3}, {2}}, V = {{1, 0}, {0, 1}})

QR(m)

Computes the \(QR\) decomposition of the matrix \(m\).

>>> QR({{12, -51, 4}, {6, 167, -68}, {-4, 24, -41}})
= (Q = {{-6/7, 69/175, 58/175}, {-3/7, -158/175, -6/175}, {2/7, -6/35, 33/35}}, R = {{-14, -21, 14}, {0, -175, 70}, {0, 0, -35}})