MATH 1554: Linear Algebra Terms
(A^T)^-1
(A^-1)^T
if A is not invertible, then an eigenvalue is
0
stochastic matrices have an eigenvalue of
1
if A is a square matrix with orthonormal columns then det A =
1 or -1
if A is invertible, then det (A^-1) =
1/det A
(A^T)^T
A
symmetric matrix
A = A^T
n x n matrices are similar if
A = PBP^-1
A is diagonalizable if it is similar to diagonal matrix D, write as
A = PDP^-1
(AB)^-1
B^-1A^-1
span
set of all linear combinations of vectors
regular stochastic matrix if
some k such that P^k only contains positive elements, where P is a stochastic matrix
stochastic matrix
square matrix whose columns are probability vectors
upper triangular
zeros located on lower left
lower triangular
zeros located upper right
vertical shear
{[1 0] [k 1]}
horizontal shear
{[1 k] [0 1]}
counterclockwise rotation standard matrix in R2
{[cos -sin] [sin cos]}
clockwise rotation standard matrix in R2
{[cos sin] [-sin cos]}
distance between vector u and v
‖u-v‖
length of a vector
‖u‖ = √u x u = √u₁² + .... +un²
if 2 rows are interchanged odd number of times to produce B, then det B =
-det A
ordering of geometric and algebraic multiplicities of eigenvalues
1 ≤g≤a
echelon form
1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.
row reduced echelon form
1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros. 4. All leading entries are 1 5. Leading entries are only nonzero entry in column
homogeneous coordinates
1s on main diagonal and 0 everywhere except for translation part on rightmost column
(A^-1)^-1
A
A is diagonalizable if
A has n linearly independent eigenvectors
eigenvector calculation
A-lamda I
invertible = non-singular
AC = CA = I
(A+B)^T
A^T + B^T
normal equations
A^TAxhat = A^T b
if eigenvector of A then
Av = lamda v
nontrivial solution
Ax = 0 has this if there is a free variable
(AB)^T
B^TA^T
google page rank equation
G = pP* + (1 - p)k --> k matrix has all elements = 1/n; p is damping factor and if p is greater than or equal to 0 but less than 1, it makes G a regular stochastic matrix
Row A is orthogonal to
Nul A
Row A perp equals
Nul A
Diagonalize to A= PDP^-1
P is eigenvectors and D are eigenvalues in order of eigenvectors
google page rank
P matrix formed from probability of going to different pages; if no link, equal probability for all pages (P*)
A=QR
Q obtained through Gram-Schmidt and R obtained through Q^TA
quadratic form
Q(x) = x^T Ax
negative definite
Q<0 for all x ≠0
positive definite
Q>0 for all x ≠0
positive semidefinite
Q≥0 for all x
linear transformations
Rn to Rm; domain of T - Rn; codomain of T - Rm
unique least-squares solution
Rxhat=Q^Tb
PCP^-1 for imaginary eigenvalues
Solve with lambda then substitute after getting eigenvector; P= (Realv Imv), C= ([a -b] [b a])
an mxn matrix has orthonormal columns if
U^TU = I; has to have linearly independent columns
LU factorization
Ux = y; solve for y in Ly = b; solve for x in Ux = y TO GET LU DECOMP: reduce A to echelon form U by sequence of row operations; place entries in L with opposite sign in same position
determinant of 2x2
ad - bc
eigenvalues of triangular matrix
are on diagonal
if A and B are similar their eigenvalues
are the same
row reductions can or cannot change eigenvalues?
can change
calculate eigenvalue
det (A-λI) = 0 roots are eigenvalues
det A^T =
det A
if a multiple of a row of A is added to another row to produce B then det B =
det A
det (AB)
det A x det B
invertibility does not tell anything about
diagonalizability
geometric multiplicity of eigenvalue
dimension of Null (A-λI)
free variables
don't have pivot in column
orthogonal definition
dot product of u and v = 0; orthogonal vectors are linearly independent
horizontal contraction/expansion
e1 changes
vertical contraction/expansion
e2 changes
equivalent statements for diagnolizability
g = a; eigenvectors of A form a basis for Rn
linearly dependent
has non-trivial solutions
2x2 matrix invertibility
if ad-bc ≠ 0, then it is invertible; inverse equals 1/ (ad - bc) times the matrix {[d -b] [-c a]}
Ax = b has a solution
if an only if b is a linear combination of the columns of A
subspace
if it is closed under scalar multiplies and vector addition
if det A ≠ 0, then A is invertible or not?
invertible
equivalent statements
invertible; row equivalent to identity; n pivotal columns; only trivial solution for Ax = 0; linearly independent columns; Ax = b has 1 solution for all b in Rn; columns of A span Rn; A^T is invertible
if A is nxn and has n distinct eigenvalyes,
it is diagonalizable
if one row of A is multiplied by a scalar k to produce B, then det B =
k det A
algebraic multiplicity of eigenvalue
multiplicity as a root of characteristic polynomial
dimension of a subspace
number of vectors in a basis of H; pivot columns
if A is symmetric with eigenvectors v1 and v2 with two different eigenvalues, then v1 and v2 are
orthgonal
orthonormal basis
orthogonal basis in which every vector u has unit length for each vector w
one-to-one transformation
pivot in every column; trivial solution if solution exists; linearly independent columns
onto transformation
pivot in every row; columns span Rm; solution for all b
basic variables
pivot position
reflection through x1 axis, reflection through x2 axis, reflection through x1 = x2 axis, reflection through x1 = -x2 axis
plot e1 (1,0) and e2 (0,1) and perform to find new
steady state vectors
probability (unit) vector q such that Pq = q, where P is stochastic matrix; (P - I) q = 0
determinant of a triangular matrix
product of entries on main diagonal
any matrix invertibility
row reduce augmented matrix (A|In) to RREF
Markov chains
sequence of probability vectors and stochastic matrix
linearly independent
set of vectors only has the trivial solution
Col A
subspace of Rm; pivotal columns of A
Nul A
subspace of Rn; Ax = 0
trace
sum of diagonal elements
if a matrix is square and diagonal then it is symmetric or not?
symmetric
if vector x is in Nul A^T
then A^Tx = 0; x is orthogonal to rows of A^T; x is orthogonal to columns of A; Col A is orthogonal to Nul A^T
if it is a regular stochastic matrix, then
there is a unique steady-state vector
standard vector
vector in Rn in which every entry is 0, except for entry i = 1
Gram-Schmidt Process
vector vp = vector xp - ((dot product of xp and v1)/(dot product of v1 and v1)) x v1; v is an orthogonal basis for W, which x is a basis for a subspace of
probability vector
vector x with non-negative elements that sum to 1
W perp
vector z is orthogonal to subspace W if z is orthogonal to every vector in W; set of all vectors orthogonal to W is a subspace
projection onto either axis
whichever axis it is projecting onto, keep values for that same, but other one becomes 0 fully
change of variable
x = Py; Q = y^TDy= eigenvalue y1^2 + eigenvalue 2 y2^2
Markov chain
x1 = Px0 = P(cv + ....)
closest vector in span of vector u is
y hat = ((dot product of vector y and u)/ (dot product of u and u)) x (u) = projection of y onto u