matrix_types [Tradejack]

Matrix types

Non-zeroes on only main diagonal, and diagonals above and below it.

Hermitian matrix always reducable to tridiagonal.

Matrix equiv of “positive”, “nonnegative”, etc.

$n \times n$ Hermitian matrix.

Equiv.: all eigenvalues are nonneg./positive/nonpos./negative.

Semidefinite positive: $x^* M x \geq 0$

Definite positive: $x^* M x > 0$

$m_{ij} = m^*_{ij}$ , where $m^*$ is complex conjugate (imaginary portion negated).

$A A^* = A^* A$ .

Spectral theory generally applies to only normal matrices.

$(A^*)_{ij} = \bar{A_{ji}}$ ; i.e. transpose A and take elementwise complex conjugates.

If A is a real matrix, $A^* = A^T$ .