• d = det(A) computes thedeterminant of the matrixA.
• lambda = eig(A) returns theeigenvalues ofA in the vectorlambda, and
• [V, lambda] = eig(A) also returns theeigenvectors inV butlambda is now a matrix whose diagonals contain the eigenvalues. This relationship holds true (within round off errors)A = V*lambda*inv(V).
• inv(A) computes the inverse of non-singular matrixA. Note that calculating the inverse is often 'not' necessary. See the next two operators as examples. Note that in theoryA*inv(A) should return the identity matrix, but in practice, there may be some round off errors so the result may not be exact.
• A / B computesX such that${\displaystyle XB=A}$. This is called right division and is done without forming the inverse ofB.
• A \ B computesX such that${\displaystyle AX=B}$. This is called left division and is done without forming the inverse ofA.
• norm(A, p) computes thep-norm of the matrix (or vector)A. The second argument is optional with default value${\displaystyle p=2}$.
• rank(A) computes the (numerical)rank of a matrix.
• trace(A) computes thetrace (sum of the diagonal elements) ofA.
• expm(A) computes the matrix exponential of a square matrix. This is defined as

${\displaystyle I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots }$

• logm(A) computes thematrix logarithm of a square matrix.
• sqrtm(A) computes thematrix square root of a square matrix.

Below are some more linear algebra functions. Usehelp to find out more about them.

• balance (eigenvalue balancing),
• cond (condition number),
• dmult (computes diag(x) * A efficiently),
• dot (dot product),
• givens (Givens rotation),
• kron (Kronecker product),
• null (orthonormal basis of the null space),
• orth (orthonormal basis of the range space),
• pinv (pseudoinverse),
• syl (solves the Sylvester equation).

• R = chol(A) computes the Cholesky factorization of the symmetric positive definite matrixA, i.e. the upper triangular matrixR such that${\displaystyle R^{T}R=A}$.
• [L, U] = lu(A) computes the LU decomposition ofA, i.e.L is lower triangular,U upper triangular and${\displaystyle A=LU}$.
• [Q, R] = qr(A) computes the QR decomposition ofA, i.e.Q is orthogonal,R is upper triangular and${\displaystyle A=QR}$.

Below are some more available factorizations. Usehelp to find out more about them.

• qz (generalized eigenvalue problem: QZ decomposition),
• qzhess (Hessenberg-triangular decomposition),
• schur (Schur decomposition),
• svd (singular value decomposition),
• housh (Householder reflections),
• krylov (Orthogonal basis of block Krylov subspace).

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