# Octave Programming Tutorial/Linear algebra

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# Functions

[edit | edit source]`d = det(A)`

computes the determinant of the matrix*A*.`lambda = eig(A)`

returns the eigenvalues of`A`

in the vector`lambda`

, and`[V, lambda] = eig(A)`

also returns the eigenvectors in`V`

but`lambda`

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 matrix*A*. Note that calculating the inverse is often 'not' necessary. See the next two operators as examples. Note that in theory`A*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`

computes*X*such that . This is called right division and is done without forming the inverse of*B*.`A \ B`

computes*X*such that . This is called left division and is done without forming the inverse of*A*.`norm(A, p)`

computes the*p*-norm of the matrix (or vector)*A*. The second argument is optional with default value .`rank(A)`

computes the (numerical) rank of a matrix.`trace(A)`

computes the trace (sum of the diagonal elements) of*A*.`expm(A)`

computes the matrix exponential of a square matrix. This is defined as

`logm(A)`

computes the matrix logarithm of a square matrix.`sqrtm(A)`

computes the matrix square root of a square matrix.

Below are some more linear algebra functions. Use `help`

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).

# Factorizations

[edit | edit source]`R = chol(A)`

computes the Cholesky factorization of the symmetric positive definite matrix*A*, i.e. the upper triangular matrix*R*such that .`[L, U] = lu(A)`

computes the LU decomposition of*A*, i.e.*L*is lower triangular,*U*upper triangular and .`[Q, R] = qr(A)`

computes the QR decomposition of*A*, i.e.*Q*is orthogonal,*R*is upper triangular and .

Below are some more available factorizations. Use `help`

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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