background

Question 1

matrix definition:

Answer 1

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

Question 2

if m=n:

Answer 2

the matrix is square, if not, it’s rectangular

Question 3

submatrix:

Answer 3

any matrix obtained by deleting rows and columns

Question 4

block matrix:

Answer 4

a partitioning of a matrix into submatrices, whose dimensions must be consistent - each element is a matrix, basically

Question 5

row vector:

Answer 5

a 1 x n matrix, e.g. [a b c d]

Question 6

column vector:

Answer 6

a n x 1 matrix, e.g. [1 / 2 / 3 / 4]

Question 7

zero matrix:

Answer 7

a matrix with all elements being 0, written O(mxn) or sometimes just O if m and n are clear from context

Question 8

identity matrix:

Answer 8

a matrix with all values being 0 except the top left to bottom right diagonal, written I(mxn) or just I

Question 9

j-th unit vector:

Answer 9

the j-th column of the identity matrix, so all 0 except the j-th row which is 1

Question 10

common notation:

Answer 10

capital letters for matrices
lower case letters for elements and vectors
lower case greek letters for scalars

Question 11

transposition:

Answer 11

switching the rows and columns
C=A^T <=> c(ij)=a(ji)

Question 12

conjugate transposition:

Answer 12

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

Question 13

properties of transposition:

Answer 13

(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

Question 14

addition:

Answer 14

C=A+B <=> c(ij)=a(ij)+b(ij)

Question 15

scalar-matrix multiplication:

Answer 15

C=αA <=> c(ij)=αa(ij)

Question 16

properties of matrix addition:

Answer 16

it’s commutative (A+B=B+A), associative ((A+B)+C=A+(B+C)), and distributive with scalar multiplication (α(A+B)=αA+αB)

Question 17

matrix-matrix multiplication:

Answer 17

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)

Question 18

properties of matrix-matrix multiplication:

Answer 18

associative (A(BC)=(AB)C)
distributive (A(B+C)=AB+AC)
NOT commutative (AB!=BA generally)

Question 19

A^0 = I if:

Answer 19

A is a nonzero square matrix

Question 20

involutory:

Answer 20

a square matrix is involutory if A^2=I

Question 21

idempotent:

Answer 21

a square matrix is idempotent if A^2=A

Question 22

nilpotent:

Answer 22

a square matrix is nilpotent if A^k=0 for some integer k>0

Question 23

if p(z) = c0 + c1z + … +ckz^k, p(A) =:

Answer 23

c0I + c1A + … + ckA^k (basically just here’s how to format polynomials of matrices)

Question 24

inner product:

Answer 24

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

Question 25

length of a vector:

Answer 25

root(x*x)

Question 26

orthogonal:

Answer 26

2 vectors are orthogonal if their inner product = 0

Question 27

orthonormal:

Answer 27

for 2 vectors that are orthogonal, they are also orthonormal if xx=yy=1

Question 28

outer product:

Answer 28

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

Question 29

diagonal matrix:

Answer 29

all elements that aren’t on the main diagonal are 0, also written D, diag(αi)

Question 30

upper triangular matrix:

Answer 30

U, has 0 elements Below the main diagonal, above is fine

Question 31

lower triangular matrix:

Answer 31

L, has 0 elements above the main diagonal

Question 32

block diagonal:

Answer 32

a block matrix with all values other than the main diagonal equal to 0

Question 33

block upper/lower triangular:

Answer 33

a block matrix with 0 elements below/above the main diagonal

Question 34

symmetric matrix:

Answer 34

A=A^T

Question 35

hermitian matrix:

Answer 35

A*=A

Question 36

orthogonal matrix:

Answer 36

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

Question 37

unitary matrix:

Answer 37

U, UU=UU=I

Question 38

permutation matrix:

Answer 38

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

Question 39

rank:

Answer 39

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

Question 40

range of a matrix:

Answer 40

the range(A) of A, A in C^(mxn), = {y in C^m: y=Ax for some x in C^n}

Question 41

null space:

Answer 41

the null(A) of A, A in C^(mxn) = {x in C^n: Ax=0}

Question 42

range of a row:

Answer 42

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

Question 43

rank of a row:

Answer 43

rank(A)=dim(range(A))
dim(V) is the maximum number of linearly independent vectors in the vector space V

Question 44

how to find n in C^(mxn):

Answer 44

rank(A)+dim(null(A))=n

Question 45

determinant of nxn matrix:

Answer 45

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

Question 46

cofactor:

Answer 46

(-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

Question 47

properties of the determinant:

Answer 47

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)

Question 48

inverse:

Answer 48

if A and B satisfy AB=I, then B is the inverse of A, B=A^-1

Question 49

singular:

Answer 49

a matrix A is nonsingular if A^-1 exists, singular if not

Question 50

properties of inverses:

Answer 50

(AB)^-1=B^(-1)A^-1
(A^-1)^T=A^-T=(A^T)^-1

Question 51

conditions that are equivalent to A being nonsingular:

Answer 51

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