Brushing up on Linear Algebra

Hermitian Matrix

Square matrix with complex entries that is equal to its own conjugate transpose.

i.e.

[3 2+i; 2-i 1]

Positive Definite Matrix

A n x n real matrix M is positive definite if z’Mz > 0 for all non-zero vector z with real # entries.

z*Mz > 0 (for complex or Hermitian Matrix M)

example:

[z_0 z_1] [1 0; 0 1] [z_0; z_1]  = z_0^2 z_1^2

Therefore, [1 0; 0 1] is positive definite

Eigenvalues

Non-zero vectors that remain parallel to the original vector no matter what matrix (read: transformation) is applied to them.

Av = lamba *v, where lambda is the eigen value of A corresponding to v.

Cholesky Decomposition

Decomposition of a Hermitian, positive-definite matrix into product of lower triangular matrix and its conjugate transpose (take the transpose then negate imaginary parts but not real part). Analogous to taking a square root of a number.

A = LL*, where L is a lower triangular matrix with positive diagonal entries.

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