# 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.