Linear Algebra Primatives Flashcards
If matrix A has an inverse, then
- the inverse is unique
- A-1A=I
- AA-1=I
(A-1)-1=
A
if A and B are non-singular matricies, then (inverse relations)
(AB)-1=B-1A-1
If A is non-singular and k ne 0, then (inverses)
(kA)-1=(1/k)A-1
If A and B are matrices such that AB is defined (ie conformable), then (transposes)
(AB)T=BTAT
A is called symmetric if
AT=A
If A and B are nxn square matrices, then (determinants)
det(AB)=det(A)det(B)
If A in nxn, then det(A) = 0 IFF
A is singular
The rank of A is
the greatest number of linearly independent columns (or rows) of A
If A and B are non-singular, then for any matrix C, (ranks)
C, AC, CB, ACB all have the same rank (assuming multiplications defined)
If A in an mxn matrix of rank r, then there exists non-singular matrices P and Q such that PAQ equals one of the following:
- I (this is “eye”)
- [I 0]
- [I 0]T
- [I 0 0 0] (square)
The rank of AB can not exceed
the rank of either the rank of A or B
If A is a nxn matrix, then det(A)=0 IFF (rank)
the rank of A is less than n
The matrix of a quadratic form can always be chosen to be
symmetric (attach proof)
A and B are said to be congruent matrices IFF
there exists a non-singular matrix, C, such that B=CTAC.
We say C is the congruent transformation of A.
Let A be an nxn symmetric of rank r. There exists a non-singular matrix C such that
CTAC = D where
D is a diagonal matrix with exactly r non-zero diagonal elements
If A and B are congruent matricies, then
they have the same rank
Let C be an mxn matrix with rank r, then the ranks of CTC and CCT are
also r.
Let A be an nxn matrix. There always exist n complex numbers …
eigenvalues (characteristic roots, symbol lambda) that satisfy
det(A-lambda I) = 0. If A is real symmetric, then all lambda’s are real.
Let A be an nxn symmetric matrix, the rank of A (in terms of eigenvalues)
is the number of non-zero eigenvalues
Let A be an nxn matrix. A has ____ eigenvalues IFF A is singular.
A has at least one zero eigenvalue.
Let A be an nxn matrix. The determinant of A (in terms of eigenvalues)
the product of its eigenvalues
Π lambdai
Let A be an nxn matrix, and let C be any nxn non-singular matrix. The following matrices all have the same eigenvalues
- A
- C-1AC
- CAC-1
Let A be an nxn real matrix. A necessary and sufficient condition that there exists a nonzero y that satisfies:
Ay=lambda * y
is that lambda be an eigenvalue of A.
Let P be an nxn matrix. P is called orthonormal IFF
P-1=P’
THUS
P’P=I
P’P=I suggests
P is orthonormal
Let A be an nxn matrix, and let P be an orthonormal matrix, then (determinants)
det(A) = det(P’AP)
Let x and y be nx1 vectors. x and y are called orthogonal if
x’y = 0
Let A be an mxn matrix and B be a nxp matrix. And and B are said to be orthoganal to each other IF
AB=0
Let A ben an nxn symmetric matrix. There exists an orthonormal matrix P such that
P’AP = D where Di is a diagonal matrix whose diagonal elements are the eigenvalues of A.
Full rank factorization. Let A be an mxn matrix with rank r>0. There exists matrices AL (mxr with rank r) and AR (rxn with rank r) such that:
A=ALAR
The column space of a matrix A is :
the set of vectors that can be generated as linear combinations of columns of A. (ditto for rowspace).
Let A be an nxn matrix. A is called positive semidefinite if:
- A=A’ (symtric)
- y’Ay ge 0 for all y
- There exists at least one y ne 0 such that y’Ay=0.
If A is positive semidefinite, then
- Rank of A is less than n
- The eigenvalues of A are greater than or equal to 0
- Let P be a nxn non-singular matrix. P’AP is also p.s.d.
Let A be an nxn matrix. A is called positive definite if
- A=A’
- y’Ay>0 for all y ne 0
If A is positive definite, then
- The rank of A is n
- All the eigenvalues of A are greater than 0
- Let P be a nxn non-singular matrix. P’AP is also positive definite
A matrix is called non-negative definite if
it is either positive definite or positive semi-definite
Let C be an mxn matrix with rank r. C’C and CC’ are both
non-negative definite. They are positive definite IFF they have full rank.
Let A be a nxn symmetric non-negative definite matrix. There exists some nxn matrix B such that
B’B=A
Let A and B be nxn symmetrix matrices. If A is positive definite, then there exists a non-singular matrix Q, such that
- Q’AQ=I and
- Q’BQ=D
where D is a diagonal matrix whose diagonal elements are roots of det(B-lambda A) = 0
If A and B are both non-negative dfinite, then there exists a matrix Q such that both Q’AQ and Q’BQ are both
diagonal
Consider a matrix of the attached form.
It can be shown that the inverse of this matrix is:
Let A be an nxn matrix. A is idempotent if
AA=A
If A is an nxn idempotent matrix with rank n, then
A=I
If A is an nxn idempotent matrix of rank less than n, then A is
positive semidefinite
If A is an nxn idempotent matrix with rank r, then it has ___ non-zero eigenvalues each equal to ___.
r non-zero eigenvalues
each equal to 1
Let A ben an nxn (symmetric) idempotent matrix.
- A’ is
- Let P be an orthonormal matrix. P’AP is
- Let P be an nxn non-singular matrrix. PAP-1 is
- I - A is
- A’ is (symetric) idempotent
- Let P be an orthonormal matrix. P’AP is (symmetric) idempotent
- Let P be an nxn non-singular matrrix. PAP-1 is idempotent
- I - A is (symmetric) idempotent
Trace A
Σ diagonal elements
Let A be mxn and B be an nxm matrix.
Trace(AB) =
trace(BA)
trace(ABC)
trace(CAB)
Let A be an nxn matrix and P be a non-singlular nxn matrix
trace(A) =
trace(P-1AP)
Let A be an nxn matrix and P be a orthonormal nxn matrix
trace(A) =
trace(A) = trace(P’AP)
Let A be an nxn matrix with eigenvalues lambda 1, … n
trace(A) =
sum of eigenvalues
If A is an idempotent matrix, then
trace(A) =
trace(A) = rank(A)
Let A and B be nxn matrices, and let a and b be scalars.
trace(aA + bB) =
trace(aA + bB) = a*trace(A) + b*trace(B)
Let A be an nxn matrix
trace(A) =
trace(A) = trace(A’)
