background Flashcards
matrix definition:
an ordered array of mn elements a(ij) - i=(1,…,m), j=(1,…,n) in the form of yknow. a matrix. with m as the row and n as the column
these elements can be taken from any field/set, but we will always be using the set of all real or all complex numbers
if m=n:
the matrix is square, if not, it’s rectangular
submatrix:
any matrix obtained by deleting rows and columns
block matrix:
a partitioning of a matrix into submatrices, whose dimensions must be consistent - each element is a matrix, basically
row vector:
a 1 x n matrix, e.g. [a b c d]
column vector:
a n x 1 matrix, e.g. [1 / 2 / 3 / 4]
zero matrix:
a matrix with all elements being 0, written O(mxn) or sometimes just O if m and n are clear from context
identity matrix:
a matrix with all values being 0 except the top left to bottom right diagonal, written I(mxn) or just I
j-th unit vector:
the j-th column of the identity matrix, so all 0 except the j-th row which is 1
common notation:
capital letters for matrices
lower case letters for elements and vectors
lower case greek letters for scalars
transposition:
switching the rows and columns
C=A^T <=> c(ij)=a(ji)
conjugate transposition:
C=A* <=> c(ij)=(a(ji)—–) (the line is over the bit in brackets with it and denotes complex conjugate)
transposes and switches the elements for their conjugates
properties of transposition:
(A^T)^T = A
(A) = A
(αA)^T = αA^T
(αA)* = (α-)A*
(A+B)^T = A^T + B^T
(A+B)* = A+B
(AB)^T = A^(T)B^(T)
(AB)* + AB
addition:
C=A+B <=> c(ij)=a(ij)+b(ij)
scalar-matrix multiplication:
C=αA <=> c(ij)=αa(ij)
properties of matrix addition:
it’s commutative (A+B=B+A), associative ((A+B)+C=A+(B+C)), and distributive with scalar multiplication (α(A+B)=αA+αB)
matrix-matrix multiplication:
C=AB <=> c(ij)=(r)Σ(k=1)a(ik)b(kj)
e.g. the first element of C is a(11)b(11) + … + a(1n)b(n1)
properties of matrix-matrix multiplication:
associative (A(BC)=(AB)C)
distributive (A(B+C)=AB+AC)
NOT commutative (AB!=BA generally)
A^0 = I if:
A is a nonzero square matrix
involutory:
a square matrix is involutory if A^2=I
idempotent:
a square matrix is idempotent if A^2=A
nilpotent:
a square matrix is nilpotent if A^k=0 for some integer k>0
if p(z) = c0 + c1z + … +ckz^k, p(A) =:
c0I + c1A + … + ckA^k (basically just here’s how to format polynomials of matrices)
inner product:
the inner product of 2 vectors x,y in the complex numbers is x*y=(n)Σ(i=1)(x(i)–)y(i)
aka the dot product, the sum of the products of corresponding components, basically means transpose 1 so when you multiply you get just a number
length of a vector:
root(x*x)
orthogonal:
2 vectors are orthogonal if their inner product = 0
orthonormal:
for 2 vectors that are orthogonal, they are also orthonormal if xx=yy=1
outer product:
xy*=[
x1(y1–) . . . x1(yn–)
… …
xm(y1–) . . . xm(yn–)]
basically just transpose one so when you multiply them you get a matrix with multiple rows and columns
diagonal matrix:
all elements that aren’t on the main diagonal are 0, also written D, diag(αi)
upper triangular matrix:
U, has 0 elements Below the main diagonal, above is fine
lower triangular matrix:
L, has 0 elements above the main diagonal
block diagonal:
a block matrix with all values other than the main diagonal equal to 0
block upper/lower triangular:
a block matrix with 0 elements below/above the main diagonal
symmetric matrix:
A=A^T
hermitian matrix:
A*=A
orthogonal matrix:
an orthogonal matrix Q satisfies QQ^T=I and Q^(T)Q=I, so that if Q=[q1,…,qn] then qi^(T)qj=δij where δij=1 if i=j and 0 otherwise
unitary matrix:
U, UU=UU=I
permutation matrix:
Pij - the identity matrix, but switch the i and jth rows
PijA swaps the i and jth rows of A, APij swaps the i and jth columns of A
Pij is orthogonal and involutory
rank:
the rank(A) of a matrix A is the maximum number of linearly independent rows or columns of A
(a set of vectors is linearly dependent if Σaivi=0 for some scalars ai not all 0, and linearly independent if not
range of a matrix:
the range(A) of A, A in C^(mxn), = {y in C^m: y=Ax for some x in C^n}
null space:
the null(A) of A, A in C^(mxn) = {x in C^n: Ax=0}
range of a row:
if A=[a1, a2, …, an], then range(A)=span{a1, a2, …, an}
span(S) denotes the set of all linear combinations of vectors in the set S
rank of a row:
rank(A)=dim(range(A))
dim(V) is the maximum number of linearly independent vectors in the vector space V
how to find n in C^(mxn):
rank(A)+dim(null(A))=n
determinant of nxn matrix:
det(A)=(n)Σ(j=1)aij(-1)^(i+j)det(Aij) for any i, for any j swap the j=1 for i=1
Aij here is a (n-1)x(n-1) matrix formed by deleting the ith row and jth column of a
cofactor:
(-1)^(i+j)det(Aij) is a cofactor of A where Aij is formed by deleting the ith row and jth column of A, is a scalar
properties of the determinant:
det(AB)=det(A)+det(B)
det(αA)=α^(n)det(A) (n is the number of columns)
if A is block diagonal or triangular with square diagonal blocks A11, A22, …, App, then det(A)=det(A11)det(A22)…det(App)
inverse:
if A and B satisfy AB=I, then B is the inverse of A, B=A^-1
singular:
a matrix A is nonsingular if A^-1 exists, singular if not
properties of inverses:
(AB)^-1=B^(-1)A^-1
(A^-1)^T=A^-T=(A^T)^-1
conditions that are equivalent to A being nonsingular:
null(A)={0} (there is no nonzero y such that Ay=0)
rank(A)=n (the rows or columns of A are linearly independent)
det(A)!=0
None of A’s eigenvalues are 0
