Eigenvalues formulas

Always valid for a 2 by 2 matrix.

A is n by n and i is any fixed row. M_ij is the determinant of the (n-1) by (n-1) matrix left after deleting row i and column j. The same works along any column. Expand along the row or column with the most zeros.

A and B are n by n. The inverse rule needs det(A) not equal to 0. Applying the product rule repeatedly gives det(A^k) = (det A)^k for every integer k >= 1.

A is n by n. The factor c comes out of each of the n rows, so the power is n, the size of the matrix.

A is n by n with n >= 2, and adj(A) is the transpose of the cofactor matrix. If det(A) is not 0, then A^{-1} = adj(A)/det(A). Also adj(adj A) = (det A)^{n-2} A for n >= 3, and adj(adj A) = A for n = 2.

In general, the n by n matrix whose i-th row is 1, x_i, x_i^2, ..., x_i^{n-1} has determinant equal to the product of (x_j - x_i) over all i < j. It is non-zero exactly when the x_i are distinct.

A is n by n. The n roots of this equation, counted with multiplicity over the complex numbers, are the eigenvalues. For each root, solve (A - lambda I)x = 0 to find the eigenvectors.

Valid for any 2 by 2 matrix; it is det(A - lambda I) = 0 written out.

c_2 is the sum of the three principal 2 by 2 minors: for each i, delete row i and column i, take the determinant, and add the three results. This is det(A - lambda I) = 0 multiplied through by -1.

A is n by n, and the eigenvalues are counted with algebraic multiplicity. Hence A is singular if and only if 0 is an eigenvalue. Applying the trace rule to A^k gives tr(A^k) = lambda_1^k + ... + lambda_n^k for every integer k >= 1.

A, P and D are n by n and P is invertible. The columns of P are n linearly independent eigenvectors, in the same order as the eigenvalues on the diagonal of D. This factorisation exists if and only if A has n linearly independent eigenvectors.

A is an n by n real symmetric matrix, Q is an n by n orthogonal matrix whose columns are orthonormal eigenvectors, and Lambda is the diagonal matrix of the matching eigenvalues. A real symmetric matrix has real eigenvalues, its eigenvectors for distinct eigenvalues are orthogonal, and it is always diagonalisable by an orthogonal matrix.

A is an n by n real symmetric matrix with smallest eigenvalue lambda_min and largest eigenvalue lambda_max. Each bound is attained at an eigenvector for that eigenvalue, so the maximum of x^T A x over unit vectors (||x|| = 1) is lambda_max and the minimum is lambda_min.

A is n by n, and the constant term c of p is replaced by cI. Every square matrix satisfies its own characteristic equation. For a 2 by 2 matrix: A^2 - tr(A) A + det(A) I = 0.