# Matrix tools

**Life cycle diagram** Constructs a diagram of the life cycle specified by a projection matrix.

**Basic analysis** Enters several array formulas that together compute some important properties of a population projection matrix – such as the generation times and eigenvalues and eigenvectors associated to the dominant eigenvalue (giving the finite rate of increase, and stable age structure and reproductive value).

**Finite rate of increase** Enters formula for the dominant eigenvalue of a population projection matrix.

**Age distribution** Enters an array formula for the right eigenvector associated to the dominant eigenvalue of a population projection matrix.

**Reproductive value** Enters an array formula for the left eigenvector associated to the dominant eigenvalue of a population projection matrix.

**Matrix sensitivity** Enters an array formula for the sensitivity of the dominant eigenvalue of a population projection matrix to small changes in the elements of the matrix.

**Matrix to row vector** Inserts a row of formulas pointing to the entries in a matrix.

**Matrix to column vector** Inserts a column of formulas pointing to the entries in a matrix.

**Matrix projection** Iterates a population projection matrix for a certain number of steps. Similar to Numerical projection but requires a fixed projection matrix and a state vector as input.

**Eigenanalysis** Enters an array formula for the eigenvalues (real and imaginary parts) and associated eigenvectors of a square matrix.

**LU decomposition** Enters an array formula that returns the lower/upper decomposition of a square matrix.

**Cholesky decomposition** Enters an array formula that returns the Cholesky decomposition of a positive definite matrix.

**QR decomposition** Enters an array formula that returns the returns the QR decomposition of a matrix with rows >= columns.

**Singular value decomposition** Enters an array formula that returns the singular value decomposition of a matrix.