Term Paper Maths

Term Paper Allotment Section-B4901 DOA: 18/09/09 Course Code:MTH101 DOS: 05/12/09 Roll No. B4901A01

B4901A02

Topic

Signature

Illustrate the definition of rank and its characteristics by column vectors with a 3X4 matrix. Show that for a square matrix the linear dependence of the row vectors implies that of the column vectors and conversely. Show that for a non square matrix either the row vectors or the column vectors must always be linearly dependent. The rank of the product of two matrices cannot exceed the rank of either factor. Illustrate this with examples. The rank r of the product of an mxn matrix A of rank and an nxp matrix B of rank satisfies + n (=the smaller of . Illustrate this with examples. Find the example in which both equality signs hold.

B4901A03

Application of Eigen values: An elastic membrane in the plane with boundary circle P:(

goes over into the point Q: (

is stretched so that a point given by y=

= AX=

. Find the principal directions, that is, the directions of the position vector X of P for which the direction of the position vector y is the same or exactly opposite. What shape does the boundary circle take under this deformation B4901A04

Prove the following statements and illustrate them with examples of your own choice.Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)If A is real, its eigen values are real or complex conjugates in pairs. (ii)

exists if and only if 0 is not an eigen value of A. It has the

eigen values

B4901A05

,

Prove the following statements and illustrate them with examples of your own choice. Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)Trace of A equals the sum of its eigen values

(ii) has the eigen values vectors as A. B4901A09

B4901A010

B4901A11

, ,

and the same eigen

Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations? Prove that the product of the two unitary matrices (of the same size) and the inverse of a unitary matrix are unitary. Give examples. Powers of unitary matrices occurring in applications may sometimes be familiar real matrices. Show that for A in

A real quadratic form Q=

and its symmetric matrix C=

are

said to be positive definite if Q>0 for all .A necessary and sufficient condition for positive definiteness is that all the determinants are positive. Show that the form 4 definite, whereas

is positive is not positive definite.

B4901A12

Find out what type of conic section the following quadratic form represents and transform it to principal axes Q=

B4901A14

Prove that if a real sequence is bounded and monotone, it converges.

B4901A15

B4901A16

Illustrate the definition of rank and its characteristics by column vectors with a 3X4 matrix. Show that for a square matrix the linear dependence of the row vectors implies that of the column vectors and conversely. Show that for a non square matrix either the row vectors or the column vectors must always be linearly dependent. The rank of the product of two matrices cannot exceed the rank of either factor. Illustrate this with examples. The rank r of the product of an mxn matrix A of rank and an nxp matrix B of rank satisfies + n (=the smaller of . Illustrate this with examples. Find the example in which both equality signs hold.

B4901A17

Application of Eigen values: An elastic membrane in the plane with boundary circle P:(

goes over into the point Q: (

is stretched so that a point given by y=

= AX=

. Find the principal directions, that is, the directions of the position vector X of P for which the direction of the position vector y is the same or exactly opposite. What shape does the boundary circle take under this deformation B4901A18

Prove the following statements and illustrate them with examples of your own choice.Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)If A is real, its eigen values are real or complex conjugates in pairs. (ii)

exists if and only if 0 is not an eigen value of A. It has the

eigen values

B4901A19

,

Prove the following statements and illustrate them with examples of your own choice. Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)Trace of A equals the sum of its eigen values (ii) has the eigen values vectors as A.

B4901A20

B4901A21

B4901A22

, ,

and the same eigen

Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations? Prove that the product of the two unitary matrices (of the same size) and the inverse of a unitary matrix are unitary. Give examples. Powers of unitary matrices occurring in applications may sometimes be familiar real matrices. Show that for A in

A real quadratic form Q=

and its symmetric matrix C=

are

said to be positive definite if Q>0 for all .A necessary and sufficient condition for positive definiteness is that all the determinants are positive. Show that the form 4 definite, whereas B4901A24

is positive is not positive definite.

Find out what type of conic section the following quadratic form

represents

B4901A27

B4901A27

B4901A28

and

transform

it

to

principal

axes

Q=

Prove that if a real sequence is bounded and monotone, it converges. Illustrate the definition of rank and its characteristics by column vectors with a 3X4 matrix. Show that for a square matrix the linear dependence of the row vectors implies that of the column vectors and conversely. Show that for a non square matrix either the row vectors or the column vectors must always be linearly dependent. The rank of the product of two matrices cannot exceed the rank of either factor. Illustrate this with examples. The rank r of the product of an mxn matrix A of rank and an nxp matrix B of rank satisfies + n (=the smaller of . Illustrate this with examples. Find the example in which both equality signs hold.

B4901A59

Application of Eigen values: An elastic membrane in the plane with boundary circle P:(

is stretched so that a point

goes over into the point Q: (

given by y=

= AX=

. Find the principal directions, that is, the directions of the position vector X of P for which the direction of the position vector y is the same or exactly opposite. What shape does the boundary circle take under this deformation B4901B30

Prove the following statements and illustrate them with examples of your own choice.Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)If A is real, its eigen values are real or complex conjugates in pairs. (ii)

exists if and only if 0 is not an eigen value of A. It has the

eigen values

B4901B32

,

Prove the following statements and illustrate them with examples of your own choice. Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)Trace of A equals the sum of its eigen values (ii) has the eigen values vectors as A.

B4901B33

, ,

and the same eigen

Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations?

B4901B37

B4901B39

Prove that the product of the two unitary matrices (of the same size) and the inverse of a unitary matrix are unitary. Give examples. Powers of unitary matrices occurring in applicationsmay sometimes be familiar real matrices. Show that for A in

A real quadratic form Q=

and its symmetric matrix C=

are

said to be positive definite if Q>0 for all .A necessary and sufficient condition for positive definiteness is that all the determinants are positive. Show that the form 4 definite, whereas

is positive is not positive definite.

B4901B40

Find out what type of conic section the following quadratic form represents and transform it to principal axes Q=

B4901B41

Prove that if a real sequence is bounded and monotone, it converges.

B4901B42

B4901B44

Illustrate the definition of rank and its characteristics by column vectors with a 3X4 matrix. Show that for a square matrix the linear dependence of the row vectors implies that of the column vectors and conversely. Show that for a non square matrix either the row vectors or the column vectors must always be linearly dependent. The rank of the product of two matrices cannot exceed the rank of either factor. Illustrate this with examples. The rank r of the product of an mxn matrix A of rank and an nxp matrix B of rank satisfies + n (=the smaller of . Illustrate this with examples. Find the example in which both equality signs hold.

B4901B45

Application of Eigen values: An elastic membrane in the plane with boundary circle P:(

goes over into the point Q: (

is stretched so that a point given by y=

= AX=

. Find the principal directions, that is, the directions of the position vector X of P for which the direction of the position vector y is the same or exactly opposite. What shape does the boundary circle take under this deformation B4901B46

Prove the following statements and illustrate them with examples of your own choice.Here, are the (not necessarily distinct)

eigen values of a given matrix A= (i)If A is real, its eigen values are real or complex conjugates inpairs. ii)

exists if and only if 0 is not an eigen value of A. It has the

eigen values

B4901B47

,

Prove the following statements and illustrate them with examples of your own choice. Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)Trace of A equals the sum of its eigen values (ii) has the eigen values vectors as A.

B4901B48

B4901B49

B4901B50

, ,

and the same eigen

Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations? Prove that the product of the two unitary matrices (of the same size) and the inverse of a unitary matrix are unitary. Give examples. Powers of unitary matrices occurring in applications may sometimes be familiar real matrices. Show that for A in

A real quadratic form Q=

and its symmetric matrix C=

are

said to be positive definite if Q>0 for all .A necessary and sufficient condition for positive definiteness is that all the determinants are positive. Show that the form 4 definite, whereas

is positive is not positive definite.

B4901B51

Find out what type of conic section the following quadratic form represents and transform it to principal axes Q=

B4901B53

Prove that if a real sequence is bounded and monotone, it converges.

B4901B55

Illustrate the definition of rank and its characteristics by column vectors with a 3X4 matrix. Show that for a square matrix the linear dependence of the row vectors implies that of the column vectors

and conversely. Show that for a non square matrix either the row vectors or the column vectors must always be linearly dependent. B4901B58

The rank of the product of two matrices cannot exceed the rank of either factor. Illustrate this with examples. The rank r of the product of an mxn matrix A of rank and an nxp matrix B of rank satisfies + n (=the smaller of . Illustrate this with examples. Find the example in which both equality signs hold. Application of Eigen values: An elastic membrane in the plane with boundary circle P:(

is stretched so that a point

goes over into the point Q: (

given by y=

= AX=

. Find the principal directions, that is, the directions of the position vector X of P for which the direction of the position vector y is the same or exactly opposite. What shape does the boundary circle take under this deformation Prove the following statements and illustrate them with examples of your own choice.Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)If A is real, its eigen values are real or complex conjugates in pairs. (ii)

exists if and only if 0 is not an eigen value of A. It has the

eigen values

,

Prove the following statements and illustrate them with examples of your own choice. Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)Trace of A equals the sum of its eigen values (ii) has the eigen values vectors as A.

, ,

and the same eigen

Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations? Prove that the product of the two unitary matrices (of the same size) and the inverse of a unitary matrix are unitary. Give examples. Powers of unitary matrices occurring in applications may sometimes be familiar real matrices. Show that for A in

A real quadratic form Q=

and its symmetric matrix C=

are

said to be positive definite if Q>0 for all .A necessary and sufficient condition for positive definiteness is that all the determinants are positive. Show that the form 4 definite, whereas

is positive is not positive definite.

Find out what type of conic section the following quadratic form represents and transform it to principal axes Q=

Prove that if a real sequence is bounded and monotone, it converges. Illustrate the definition of rank and its characteristics by column vectors with a 3X4 matrix. Show that for a square matrix the linear dependence of the row vectors implies that of the column vectors and conversely. Show that for a non square matrix either the row vectors or the column vectors must always be linearly dependent. The rank of the product of two matrices cannot exceed the rank of either factor. Illustrate this with examples. The rank r of the product of an mxn matrix A of rank and an nxp matrix B of rank satisfies + n (=the smaller of . Illustrate this with examples. Find the example in which both equality signs hold. Application of Eigen values: An elastic membrane in the plane with boundary circle P:(

goes over into the point Q: (

is stretched so that a point given by y=

= AX=

. Find the principal directions, that is, the directions of the position vector X of P for which the direction of the position vector y is the same or exactly opposite. What shape does the boundary circle take under this deformation Prove the following statements and illustrate them with examples of your own choice.Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)If A is real, its eigen values are real or complex conjugates in pairs.

(ii)

exists if and only if 0 is not an eigen value of A. It has the

eigen values

,

Prove the following statements and illustrate them withexamples of your own choice. Here, are the (not necessarily distinct) eigen values of a given matrix A= (i)Trace of A equals the sum of its eigen values (ii) has the eigen values vectors as A.

, ,

and the same eigen

Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations? Prove that the product of the two unitary matrices (of the same size) and the inverse of a unitary matrix are unitary. Give examples. Powers of unitary matrices occurring in applications may sometimes be familiar real matrices. Show that for A in