Matrix Cheat Sheet, Assignments of Acting

Download Matrix Cheat Sheet and more Assignments Acting in PDF only on Docsity! Vectors and Linear Transformations A vector space V is a set of things called basis vectors and some rules for making linear combinations of them: ax+by is a vector if x, y are vectors and a,b are numbers. A linear transformation L is a map from one vector space to another that obeys the superposition principle: L(ax+by) = aLx + bLy Every linear transformation can be represented by a matrix acting on a column vector and vice versa. This is important. An inner product maps two vectors to a number. The usual example is but others exist. The inner product of a vector with itself denes a norm. !x|y" x! 1y1 + x! 2y2 + · · · Hermitian / Symmetric Hermitian matrices are self-adjoint: . Real symmetric square matrices are Hermitian. Eigenvalues of H are real (but might be degenerate!). Eigenvectors of H form an orthogonal basis for V. (Eigenvectors corresponding to the same eigenvalue are not unique, but it is always possible to choose orthogonal ones.) A real linear combination of Hermitian matrices is Hermitian. H† = H Unitary / Orthogonal Unitary matrices obey . Real unitary matrices are orthogonal. U matrices preserve the usual inner product: . Each eigenvalue of U and the determinant of U must have complex magnitude 1. The columns of U form an orthonormal basis for V (and so do the rows) if and only if U is unitary. Two matrices L and M are similar if for some unitary U. Every rotation and/or parity transformation between two orthonormal bases is represented by a U and vice versa. U"1 = U† !Ux|Uy" = !x|y" M = ULU"1 Eigensystems and the Spectral Theorem A normal matrix N satises . Every normal matrix is similar to a diagonal matrix: where U is unitary and D is diagonal. The elements of D are eigenvalues and the columns of U are eigenvectors of N . D is unique except that the order of eigenvalues is arbitrary. is an eigenvector of N with eigenvalue if and only if . The spectrum of N (the set of its eigenvalues) can be found by solving , the characteristic polynomial of N. The product of all eigenvalues of N is and the sum of eigenvalues is , the trace of N (the sum of its diagonal elements). Two similar matrices L and M have the same spectrum, determinant, and trace (but the converse is not true). NN† = N†N N = UDU"1 Nvj = !jvj!jvj det[N # !1] = 0 det[N ] tr[N ] Misc. Terminology A matrix P is idempotent if PP = P. An idempotent Hermitian matrix is a projection. A positive-denite matrix has only positive real eigenvalues. Z is nilpotent if for some number n. The commutator of L and M is [L,M] = LM - ML .Zn = 0 Matrix Cheat Sheet Matrix Arithmetic To multiply two matrices AB, do this: (Note: a column vector is just a n x 1 matrix.) (AB)x produces the same vector as “do B, then do A to x.” Matrices add component-wise, and . To transpose M, swap its rows and columns: An (anti) symmetric matrix equals its (minus) transpose. The adjoint of M is its conjugate transpose: . Adjoint matrices obey the rule . The inverse M-1 has determinant (det[M])-1 if det[M] ! 0 . A singular matrix has determinant 0 and can"t be inverted. Transposes, adjoints and inverses obey a “backwards” rule: [AB]ij = ! k AikBkj (A + B)x = Ax + Bx [MT ]ij = Mji [M†]ij = M! ji !x|My" = !M†x|y" (AB)"1 = B"1A"1 (AB)T = BT AT (AB)† = B†A† Matrix Exponentials The exponential map of a matrix M is . The solution to the differential equation is . EXP has some, but not all, of the properties of the function : in general: only if M and N commute: only if N is invertible: . (eM )"1 = e"M (eM )T = eMT (eM )† = eM† e(a+b)M = eaMebM det[eM ] = etr[M ] eNMN!1 = NeMN"1eM+N = eMeN eNMe"N = M EXP[M ] = 1 + M + 1 2!M 2 + · · · + 1 k!M k + · · · exd dtx(t) = Mx(t) x(t) = EXP[Mt] · x(0)