Eigenvalues and Eigenvectors

Some Notations: Scalars – greek letters or italicized letters

Eigenvalues and Eigenvectors Definition. Let A be an n x n real matrix. A scalar is called an eigenvalue (characteristic root) of A if and only if there is an The vector x is called an eigenvector (characteristic vector) of A associated with α.

Example:

Theorem. The eigenvalues of a triangular matrix are its diagonal entries

Real Symmetric Matrices Even if the eigenvalues of A are not distinct it is possible, in some cases, for A to have linearly independent eigenvectors. The case A = I is an example. The identity matrix belongs to a class of matrices that have this property. This is the class of symmetric matrices. Symmetric matrices appear in the study of quadratic forms. They also turn up in optimization theory. Definition. A square matrix A is symmetric if and only if AT = A.

Quadratic Forms A quadratic form is an expression of the form

Positive vs negative semidefinite The quadratic form is said to be positive semi-definite if and only if The quadratic form is said to be negative semi-definite if and only if The matrix A is also said to be positive semidefinite (negative semidefinite).

Theorems. A real symmetric matrix is positive (negative) semidefinite if and only if its eigenvalues are nonnegative (nonpositive). A real symmetric matrix is positive (negative) definite if and only if its eigenvalues are positive (negative). (iii) A real symmetric matrix is indefinite if and only if it has positive and negative eigenvalues. (Proofs in Danao’s book) Theorem. A positive (negative) semidefinite matrix has nonnegative (nonpositive) diagonal entries. A positive (negative) definite matrix has positive (negative) diagonal entries.

Definitions. Let A be an n x n matrix and let J be a subset of the set of indices {1,2,…,n} consisting of r elements. Principal submatrix of order r denoted by PSr(A). - matrix obtained by deleting the rows and columns of A corresponding to the indices not in J. Principal minor - the determinant of PSr(A) Leading principal submatrix denoted by LPSr(A) -- The principal submatrix of order r that is obtained by deleting the rows and columns of A corresponding to the indices greater than r. Its determinant is called a leading principal minor.

1.12.10 Theorems. (i) The transposes of the principal submatrices of A are the principal submatrices of AT. (ii) The transposes of the leading principal submatrices of A are the leading principal submatrices of AT. 1.12.11. Theorem. (i) A and AT have identical principal minors. (ii) A and AT have identical leading principal minors. Proof: Exercise.

Positive semidefinite matrices have nonnegative principal minors while Negative semidefinite matrices have principal minors that alternate in sign, with principal minors of order 1 being nonpositive, principal minors of order 2 being nonnegative, and so on.