Terms and notation that you might encounter in your future studies that we
did not cover:
• Regular matrix (common use) = invertible matrix
• Singular matrix (common use) = not invertible matrix
• “∀x . . . “ = “for all values of x. . . ”
• “∃x . . . ” = “there exists a value of x. . . ”
• Sn = all permutations on n elements
• Quadratic form = a function f : Rn → R defined by f (x) = xT Ax for
some A symmetric matrix.
• Positive definite matrix = n × n symmetric matrix such that xT Ax > 0
for all x 6= 0.
• Positive semidefinite matrix = n×n symmetric matrix such that xT Ax ≥ 0
for all x ∈ Rn .
• Negative definite matrix = n × n symmetric matrix such that xT Ax < 0
for all x 6= 0.
• Negative semidefinite matrix = n×n symmetric matrix such that xT Ax ≤
0 for all x ∈ Rn .
• Indefinite matrix = n × n matrix A that is not positive semidefinite, nor
negative semidefinite.
• Unitary matrix = a complex n×n matrix U such that U −1 = U T (complex
analogue of orthogonal matrices).
• GL(n) = all invertible n×n matrices together with the operation of matrix
product (an example of a mathematical structure called a group).
• |A| = det(A)
• kAk is a norm of a matrix. Roughly speaking, a norm is a nonnegative
number that tells us how much A differs from the zero matrix. There are
several possible norms one can define. One example is the norm defined
by kAk = max{kAxk : kxk = 1}.
• Linear form = any linear transformation f : V → R.
• “Vector space over the field Zp ” = Instead of calculating in R (or C) we
will use integers {0, 1, . . . , p − 1} and do arithmetics modulo p, taking
remainders after dividing by p (ie. “4 + 2 = 1 (mod 5)”). If p is a prime
number, things work more or less like in R. The field Z2 is often used in
computer science.
• Singular value/Cholesky/LU/. . . decomposition = Decompositions often
show up in computer algorithms. I’m omitting the details here, but the
general idea is similar to QR factorization: Given a matrix A, we want to
write A as a product of “nice” matrices.
• Spectrum of A = the list of all eigenvalues of A with their algebraic multiplicities.
• Dominant eigenvalue of A = the eigenvalue of A with the largest modulus
(absolute value).
• Spectral radius of A = the maximum modulus (absolute value) of an
eigenvalue of A.
• Exponential of a matrix = Given A square matrix, we define
exp(A) =
∞
X
Ak
k=0
k!
.
This series converges for all matrices and has nice properties that are
useful when solving differential equations. For example:
d exp(At)
= A exp(At).
dt
• “Conjugate transpose”/“Hermitian conjugate”/“Hermitian adjoint”/
“bedaggered matrix”/“adjoint matrix”/“transjugate” of A = the matrix
AT (denoted by AH , A† or even A? ).
• Homomorphism of vector spaces f : V → U = Here U, V are vector spaces
and f is a mapping that satisfies f (0) = 0, f (g + h) = f (g) + f (h) for any
g, h ∈ V and f (ch) = cf (h) for any c ∈ R (or C).