Linear Algebra Primatives

Question 1

If matrix A has an inverse, then

Answer 1

• the inverse is unique
• A^-1A=I
• AA^-1=I

Question 2

(A^-1)^-1=

Answer 2

A

Question 3

if A and B are non-singular matricies, then (inverse relations)

Answer 3

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

Question 4

If A is non-singular and k ne 0, then (inverses)

Answer 4

(kA)^-1=(1/k)A^-1

Question 5

If A and B are matrices such that AB is defined (ie conformable), then (transposes)

Answer 5

(AB)^T=B^TA^T

Question 6

A is called symmetric if

Answer 6

A^T=A

Question 7

If A and B are nxn square matrices, then (determinants)

Answer 7

det(AB)=det(A)det(B)

Question 8

If A in nxn, then det(A) = 0 IFF

Answer 8

A is singular

Question 9

The rank of A is

Answer 9

the greatest number of linearly independent columns (or rows) of A

Question 10

If A and B are non-singular, then for any matrix C, (ranks)

Answer 10

C, AC, CB, ACB all have the same rank (assuming multiplications defined)

Question 11

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:

Answer 11

• I (this is “eye”)
• [I 0]
• [I 0]^T
• [I 0 0 0] (square)

Question 12

The rank of AB can not exceed

Answer 12

the rank of either the rank of A or B

Question 13

If A is a nxn matrix, then det(A)=0 IFF (rank)

Answer 13

the rank of A is less than n

Question 14

The matrix of a quadratic form can always be chosen to be

Answer 14

symmetric (attach proof)

Question 15

A and B are said to be congruent matrices IFF

Answer 15

there exists a non-singular matrix, C, such that B=C^TAC.

We say C is the congruent transformation of A.

Question 16

Let A be an nxn symmetric of rank r. There exists a non-singular matrix C such that

Answer 16

C^TAC = D where

D is a diagonal matrix with exactly r non-zero diagonal elements

Question 17

If A and B are congruent matricies, then

Answer 17

they have the same rank

Question 18

Let C be an mxn matrix with rank r, then the ranks of C^TC and CC^T are

Answer 18

also r.

Question 19

Let A be an nxn matrix. There always exist n complex numbers …

Answer 19

eigenvalues (characteristic roots, symbol lambda) that satisfy

det(A-lambda I) = 0. If A is real symmetric, then all lambda’s are real.

Question 20

Let A be an nxn symmetric matrix, the rank of A (in terms of eigenvalues)

Answer 20

is the number of non-zero eigenvalues

Question 21

Let A be an nxn matrix. A has ____ eigenvalues IFF A is singular.

Answer 21

A has at least one zero eigenvalue.

Question 22

Let A be an nxn matrix. The determinant of A (in terms of eigenvalues)

Answer 22

the product of its eigenvalues

Π lambda_i

Question 23

Let A be an nxn matrix, and let C be any nxn non-singular matrix. The following matrices all have the same eigenvalues

Answer 23

• A
• C^-1AC
• CAC^-1

Question 24

Let A be an nxn real matrix. A necessary and sufficient condition that there exists a nonzero y that satisfies:

Ay=lambda * y

Answer 24

is that lambda be an eigenvalue of A.

Question 25

Let P be an nxn matrix. P is called orthonormal IFF

Answer 25

P^-1=P’

THUS

P’P=I

Question 26

P’P=I suggests

Answer 26

P is orthonormal

Question 27

Let A be an nxn matrix, and let P be an orthonormal matrix, then (determinants)

Answer 27

det(A) = det(P’AP)

Question 28

Let x and y be nx1 vectors. x and y are called orthogonal if

Answer 28

x’y = 0

Question 29

Let A be an mxn matrix and B be a nxp matrix. And and B are said to be orthoganal to each other IF

Answer 29

AB=0

Question 30

Let A ben an nxn symmetric matrix. There exists an orthonormal matrix P such that

Answer 30

P’AP = D where Di is a diagonal matrix whose diagonal elements are the eigenvalues of A.

Question 31

Full rank factorization. Let A be an mxn matrix with rank r>0. There exists matrices A_L (mxr with rank r) and A_R (rxn with rank r) such that:

Answer 31

A=A_LA_R

Question 32

The column space of a matrix A is :

Answer 32

the set of vectors that can be generated as linear combinations of columns of A. (ditto for rowspace).

Question 33

Let A be an nxn matrix. A is called positive semidefinite if:

Answer 33

• A=A’ (symtric)
• y’Ay ge 0 for all y
• There exists at least one y ne 0 such that y’Ay=0.

Question 34

If A is positive semidefinite, then

Answer 34

• 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.

Question 35

Let A be an nxn matrix. A is called positive definite if

Answer 35

• A=A’
• y’Ay>0 for all y ne 0

Question 36

If A is positive definite, then

Answer 36

• 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

Question 37

A matrix is called non-negative definite if

Answer 37

it is either positive definite or positive semi-definite

Question 38

Let C be an mxn matrix with rank r. C’C and CC’ are both

Answer 38

non-negative definite. They are positive definite IFF they have full rank.

Question 39

Let A be a nxn symmetric non-negative definite matrix. There exists some nxn matrix B such that

Answer 39

B’B=A

Question 40

Let A and B be nxn symmetrix matrices. If A is positive definite, then there exists a non-singular matrix Q, such that

Answer 40

• Q’AQ=I and
• Q’BQ=D

where D is a diagonal matrix whose diagonal elements are roots of det(B-lambda A) = 0

Question 41

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

Answer 41

diagonal

Question 42

Answer 42

Question 43

Consider a matrix of the attached form.

It can be shown that the inverse of this matrix is:

Answer 43

Question 44

Let A be an nxn matrix. A is idempotent if

Answer 44

AA=A

Question 45

If A is an nxn idempotent matrix with rank n, then

Answer 45

A=I

Question 46

If A is an nxn idempotent matrix of rank less than n, then A is

Answer 46

positive semidefinite

Question 47

If A is an nxn idempotent matrix with rank r, then it has ___ non-zero eigenvalues each equal to ___.

Answer 47

r non-zero eigenvalues

each equal to 1

Question 48

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

Answer 48

• 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

Question 49

Trace A

Answer 49

Σ diagonal elements

Question 50

Let A be mxn and B be an nxm matrix.

Trace(AB) =

Answer 50

trace(BA)

Question 51

trace(ABC)

Answer 51

trace(CAB)

Question 52

Let A be an nxn matrix and P be a non-singlular nxn matrix

trace(A) =

Answer 52

trace(P^-1AP)

Question 53

Let A be an nxn matrix and P be a orthonormal nxn matrix

trace(A) =

Answer 53

trace(A) = trace(P’AP)

Question 54

Let A be an nxn matrix with eigenvalues lambda 1, … n

trace(A) =

Answer 54

sum of eigenvalues

Question 55

If A is an idempotent matrix, then

trace(A) =

Answer 55

trace(A) = rank(A)

Question 56

Let A and B be nxn matrices, and let a and b be scalars.

trace(aA + bB) =

Answer 56

trace(aA + bB) = a*trace(A) + b*trace(B)

Question 57

Let A be an nxn matrix

trace(A) =

Answer 57

trace(A) = trace(A’)