Vectors And Matrices
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Paper 1, Section I, B
commentThe matrix
represents a linear map with respect to the bases
Find the matrix that represents with respect to the bases
Paper 1, Section I, C
comment(a) Find all complex solutions to the equation .
(b) Write down an equation for the numbers which describe, in the complex plane, a circle with radius 5 centred at . Find the points on the circle at which it intersects the line passing through and .
Paper 1, Section II, 8B
comment(a) Consider the matrix
Find the kernel of for each real value of the constant . Hence find how many solutions there are to
depending on the value of . [There is no need to find expressions for the solution(s).]
(b) Consider the reflection map defined as
where is a unit vector normal to the plane of reflection.
(i) Find the matrix which corresponds to the map in terms of the components of .
(ii) Prove that a reflection in a plane with unit normal followed by a reflection in a plane with unit normal vector (both containing the origin) is equivalent to a rotation along the line of intersection of the planes with an angle twice that between the planes.
[Hint: Choose your coordinate axes carefully.]
(iii) Briefly explain why a rotation followed by a reflection or vice-versa can never be equivalent to another rotation.
Part IA, 2021 List of Questions
Paper 1, Section II, A
commentLet be a real, symmetric matrix.
We say that is positive semi-definite if for all . Prove that is positive semi-definite if and only if all the eigenvalues of are non-negative. [You may quote results from the course, provided that they are clearly stated.]
We say that has a principal square root if for some symmetric, positive semi-definite matrix . If such a exists we write . Show that if is positive semi-definite then exists.
Let be a real, non-singular matrix. Show that is symmetric and positive semi-definite. Deduce that exists and is non-singular. By considering the matrix
or otherwise, show for some orthogonal matrix and a symmetric, positive semi-definite matrix .
Describe the transformation geometrically in the case .
Paper 1, Section II, A
comment(a) For an matrix define the characteristic polynomial and the characteristic equation.
The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. Verify this in the case .
(b) Define the adjugate matrix of an matrix in terms of the minors of . You may assume that
where is the identity matrix. Show that if and are non-singular matrices then
(c) Let be an arbitrary matrix. Explain why
(i) there is an such that is non-singular for ;
(ii) the entries of are polynomials in .
Using parts (i) and (ii), or otherwise, show that holds for all matrices .
(d) The characteristic polynomial of the arbitrary matrix is
By considering adj , or otherwise, show that
[You may assume the Cayley-Hamilton theorem.]
Paper 1, Section II, C
commentUsing the standard formula relating products of the Levi-Civita symbol to products of the Kronecker , prove
Define the scalar triple product of three vectors , and in in terms of the dot and cross product. Show that
Given a basis for which is not necessarily orthonormal, let
Show that is also a basis for . [You may assume that three linearly independent vectors in form a basis.]
The vectors are constructed from in the same way that , are constructed from . Show that
An infinite lattice consists of all points with position vectors given by
Find all points with position vectors such that is an integer for all integers , .
Paper 1, Section I, C
commentGiven a non-zero complex number , where and are real, find expressions for the real and imaginary parts of the following functions of in terms of and :
(i) ,
(ii)
(iii) ,
(iv) ,
where is the complex conjugate of .
Now assume and find expressions for the real and imaginary parts of all solutions to
(v) .
Paper 1, Section II,
commentWhat does it mean to say an matrix is Hermitian?
What does it mean to say an matrix is unitary?
Show that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal.
Suppose that is an Hermitian matrix with distinct eigenvalues and corresponding normalised eigenvectors . Let denote the matrix whose columns are . Show directly that is unitary and , where is a diagonal matrix you should specify.
If is unitary and diagonal, must it be the case that is Hermitian? Give a proof or counterexample.
Find a unitary matrix and a diagonal matrix such that
Paper 1, Section II, C
comment(a) Let , and be three distinct points in the plane which are not collinear, and let , and be their position vectors.
Show that the set of points in equidistant from and is given by an equation of the form
where is a unit vector and is a scalar, to be determined. Show that is perpendicular to .
Show that if satisfies
then
How do you interpret this result geometrically?
(b) Let and be constant vectors in . Explain why the vectors satisfying
describe a line in . Find an expression for the shortest distance between two lines , where .
Paper 1, Section I,
comment(a) If
where , what is the value of ?
(b) Evaluate
(c) Find a complex number such that
(d) Interpret geometrically the curve defined by the set of points satisfying
in the complex -plane.
Paper 1, Section I, A
commentIf is an by matrix, define its determinant .
Find the following in terms of and a scalar , clearly showing your argument:
(i) , where is obtained from by multiplying one row by .
(ii) .
(iii) , where is obtained from by switching row and row .
(iv) , where is obtained from by adding times column to column .
Paper 1, Section II,
commentLet be the standard basis vectors of . A second set of vectors are defined with respect to the standard basis by
The are the elements of the matrix . State the condition on under which the set forms a basis of .
Define the matrix that, for a given linear transformation , gives the relation between the components of any vector and those of the corresponding , with the components specified with respect to the standard basis.
Show that the relation between the matrix and the matrix of the same transformation with respect to the second basis is
Consider the matrix
Find a matrix such that is diagonal. Give the elements of and demonstrate explicitly that the relation between and holds.
Give the elements of for any positive integer .
Paper 1, Section II, 7B
comment(a) Let be an matrix. Define the characteristic polynomial of . [Choose a sign convention such that the coefficient of in the polynomial is equal to State and justify the relation between the characteristic polynomial and the eigenvalues of . Why does have at least one eigenvalue?
(b) Assume that has distinct eigenvalues. Show that . [Each term in corresponds to a term in
(c) For a general matrix and integer , show that , where Hint: You may find it helpful to note the factorization of .]
Prove that if has an eigenvalue then has an eigenvalue where .
Paper 1, Section II, A
commentThe exponential of a square matrix is defined as
where is the identity matrix. [You do not have to consider issues of convergence.]
(a) Calculate the elements of and , where
and is a real number.
(b) Show that and that
(c) Consider the matrices
Calculate:
(i) ,
(ii) .
(d) Defining
find the elements of the following matrices, where is a natural number:
(i)
(ii)
[Your answers to parts and should be in closed form, i.e. not given as series.]
Paper 1, Section II, C
comment(a) Use index notation to prove .
Hence simplify
(i) ,
(ii) .
(b) Give the general solution for and of the simultaneous equations
Show in particular that and must lie at opposite ends of a diameter of a sphere whose centre and radius should be found.
(c) If two pairs of opposite edges of a tetrahedron are perpendicular, show that the third pair are also perpendicular to each other. Show also that the sum of the lengths squared of two opposite edges is the same for each pair.
Paper 1, Section I, A
commentThe map is defined for , where is a unit vector in and is a real constant.
(i) Find the values of for which the inverse map exists, as well as the inverse map itself in these cases.
(ii) When is not invertible, find its image and kernel. What is the value of the rank and the value of the nullity of ?
(iii) Let . Find the components of the matrix such that . When is invertible, find the components of the matrix such that .
Paper 1, Section I, C
commentFor define the principal value of . State de Moivre's theorem.
Hence solve the equations (i) , (ii) , (iii) (iv)
[In each expression, the principal value is to be taken.]
Paper 1, Section II,
commentLet be non-zero real vectors. Define the inner product in terms of the components and , and define the norm . Prove that . When does equality hold? Express the angle between and in terms of their inner product.
Use suffix notation to expand .
Let be given unit vectors in , and let . Obtain expressions for the angle between and each of and , in terms of and . Calculate for the particular case when the angles between and are all equal to , and check your result for an example with and an example with .
Consider three planes in passing through the points and , respectively, with unit normals and , respectively. State a condition that must be satisfied for the three planes to intersect at a single point, and find the intersection point.
Paper 1, Section II, A
commentWhat is the definition of an orthogonal matrix ?
Write down a matrix representing the rotation of a 2-dimensional vector by an angle around the origin. Show that is indeed orthogonal.
Take a matrix
where are real. Suppose that the matrix is diagonal. Determine all possible values of .
Show that the diagonal entries of are the eigenvalues of and express them in terms of the determinant and trace of .
Using the above results, or otherwise, find the elements of the matrix
as a function of , where is a natural number.
Paper 1, Section II, B
commentLet be a real symmetric matrix.
(a) Prove the following:
(i) Each eigenvalue of is real and there is a corresponding real eigenvector.
(ii) Eigenvectors corresponding to different eigenvalues are orthogonal.
(iii) If there are distinct eigenvalues then the matrix is diagonalisable.
Assuming that has distinct eigenvalues, explain briefly how to choose (up to an arbitrary scalar factor) the vector such that is maximised.
(b) A scalar and a non-zero vector such that
are called, for a specified matrix , respectively a generalised eigenvalue and a generalised eigenvector of .
Assume the matrix is real, symmetric and positive definite (i.e. for all non-zero complex vectors ).
Prove the following:
(i) If is a generalised eigenvalue of then it is a root of .
(ii) Each generalised eigenvalue of is real and there is a corresponding real generalised eigenvector.
(iii) Two generalised eigenvectors , corresponding to different generalised eigenvalues, are orthogonal in the sense that .
(c) Find, up to an arbitrary scalar factor, the vector such that the value of is maximised, and the corresponding value of , where
Paper 1, Section II, B
comment(a) Consider the matrix
representing a rotation about the -axis through an angle .
Show that has three eigenvalues in each with modulus 1 , of which one is real and two are complex (in general), and give the relation of the real eigenvector and the two complex eigenvalues to the properties of the rotation.
Now consider the rotation composed of a rotation by angle about the -axis followed by a rotation by angle about the -axis. Determine the rotation axis and the magnitude of the angle of rotation .
(b) A surface in is given by
By considering a suitable eigenvalue problem, show that the surface is an ellipsoid, find the lengths of its semi-axes and find the position of the two points on the surface that are closest to the origin.
Paper 1, Section I, A
commentConsider with and , where .
(a) Prove algebraically that the modulus of is and that the argument is . Obtain these results geometrically using the Argand diagram.
(b) Obtain corresponding results algebraically and geometrically for .
Paper 1, Section I, C
commentLet and be real matrices.
Show that .
For any square matrix, the matrix exponential is defined by the series
Show that . [You are not required to consider issues of convergence.]
Calculate, in terms of and , the matrices and in the series for the matrix product
Hence obtain a relation between and which necessarily holds if is an orthogonal matrix.
Paper 1, Section II,
comment(a) Given consider the linear transformation which maps
Express as a matrix with respect to the standard basis , and determine the rank and the dimension of the kernel of for the cases (i) , where is a fixed number, and (ii) .
(b) Given that the equation
where
has a solution, show that .
Paper 1, Section II, A
comment(a) Define the vector product of the vectors and in . Use suffix notation to prove that
(b) The vectors are defined by , where and are fixed vectors with and , and is a positive constant.
(i) Write as a linear combination of and . Further, for , express in terms of and . Show, for , that .
(ii) Let be the point with position vector . Show that lie on a pair of straight lines.
(iii) Show that the line segment is perpendicular to . Deduce that is parallel to .
Show that as if , and give a sketch to illustrate the case .
(iv) The straight line through the points and makes an angle with the straight line through the points and . Find in terms of .
Paper 1, Section II, B
comment(a) Show that a square matrix is anti-symmetric if and only if for every vector .
(b) Let be a real anti-symmetric matrix. Show that the eigenvalues of are imaginary or zero, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal (in the sense that , where the dagger denotes the hermitian conjugate).
(c) Let be a non-zero real anti-symmetric matrix. Show that there is a real non-zero vector a such that .
Now let be a real vector orthogonal to . Show that for some real number .
The matrix is defined by the exponential series Express and in terms of and .
[You are not required to consider issues of convergence.]
Paper 1, Section II, B
comment(a) Show that the eigenvalues of any real square matrix are the same as the eigenvalues of .
The eigenvalues of are and the eigenvalues of are , . Determine, by means of a proof or a counterexample, whether the following are necessary valid: (i) ; (ii) .
(b) The matrix is given by
where and are orthogonal real unit vectors and is the identity matrix.
(i) Show that is an eigenvector of , and write down a linearly independent eigenvector. Find the eigenvalues of and determine whether is diagonalisable.
(ii) Find the eigenvectors and eigenvalues of .
Paper 1, Section I, A
commentLet be a solution of
where and . For which values of do the following hold?
(i) .
(ii) .
(iii) .
Paper 1, Section I, C
commentWrite down the general form of a rotation matrix. Let be a real matrix with positive determinant such that for all . Show that is a rotation matrix.
Let
Show that any real matrix which satisfies can be written as , where is a real number and is a rotation matrix.
Paper 1, Section II,
comment(a) Show that the equations
determine and uniquely if and only if .
Write the following system of equations
in matrix form . Use Gaussian elimination to solve the system for , and . State a relationship between the rank and the kernel of a matrix. What is the rank and what is the kernel of ?
For which values of , and is it possible to solve the above system for and ?
(b) Define a unitary matrix. Let be a real symmetric matrix, and let be the identity matrix. Show that for arbitrary , where . Find a similar expression for . Prove that is well-defined and is a unitary matrix.
Paper 1, Section II,
commentThe real symmetric matrix has eigenvectors of unit length , with corresponding eigenvalues , where . Prove that the eigenvalues are real and that .
Let be any (real) unit vector. Show that
What can be said about if
Let be the set of all (real) unit vectors of the form
Show that for some . Deduce that
The matrix is obtained by removing the first row and the first column of . Let be the greatest eigenvalue of . Show that
Paper 1, Section II, A
comment(a) Use suffix notation to prove that
(b) Show that the equation of the plane through three non-colinear points with position vectors and is
where is the position vector of a point in this plane.
Find a unit vector normal to the plane in the case and .
(c) Let be the position vector of a point in a given plane. The plane is a distance from the origin and has unit normal vector , where . Write down the equation of this plane.
This plane intersects the sphere with centre at and radius in a circle with centre at and radius . Show that
Find in terms of and . Hence find in terms of and .
Paper 1, Section II, B
commentWhat does it mean to say that a matrix can be diagonalised? Given that the real matrix has eigenvectors satisfying , explain how to obtain the diagonal form of . Prove that is indeed diagonal. Obtain, with proof, an expression for the trace of in terms of its eigenvalues.
The elements of are given by
Determine the elements of and hence show that, if is an eigenvalue of , then
Assuming that can be diagonalised, give its diagonal form.
Paper 1, Section I,
commentPrecisely one of the four matrices specified below is not orthogonal. Which is it?
Give a brief justification.
Given that the four matrices represent transformations of corresponding (in no particular order) to a rotation, a reflection, a combination of a rotation and a reflection, and none of these, identify each matrix. Explain your reasoning.
[Hint: For two of the matrices, and say, you may find it helpful to calculate and , where is the identity matrix.]
Paper 1, Section I, B
comment(a) Describe geometrically the curve
where and are positive, distinct, real constants.
(b) Let be a real number not equal to an integer multiple of . Show that
and derive a similar expression for .
Paper 1, Section II,
comment(i) Consider the map from to represented by the matrix
where . Find the image and kernel of the map for each value of .
(ii) Show that any linear map may be written in the form for some fixed vector . Show further that is uniquely determined by .
It is given that and that the vectors
lie in the kernel of . Determine the set of possible values of a.
Paper 1, Section II, 5B
comment(i) State and prove the Cauchy-Schwarz inequality for vectors in . Deduce the inequalities
for .
(ii) Show that every point on the intersection of the planes
where , satisfies
What happens if
(iii) Using your results from part (i), or otherwise, show that for any ,
Paper 1, Section II, A
comment(a) A matrix is called normal if . Let be a normal complex matrix.
(i) Show that for any vector ,
(ii) Show that is also normal for any , where denotes the identity matrix.
(iii) Show that if is an eigenvector of with respect to the eigenvalue , then is also an eigenvector of , and determine the corresponding eigenvalue.
(iv) Show that if and are eigenvectors of with respect to distinct eigenvalues and respectively, then and are orthogonal.
(v) Show that if has a basis of eigenvectors, then can be diagonalised using an orthonormal basis. Justify your answer.
[You may use standard results provided that they are clearly stated.]
(b) Show that any matrix satisfying is normal, and deduce using results from (a) that its eigenvalues are real.
(c) Show that any matrix satisfying is normal, and deduce using results from (a) that its eigenvalues are purely imaginary.
(d) Show that any matrix satisfying is normal, and deduce using results from (a) that its eigenvalues have unit modulus.
Paper 1, Section II, A
comment(i) Find the eigenvalues and eigenvectors of the following matrices and show that both are diagonalisable:
(ii) Show that, if two real matrices can both be diagonalised using the same basis transformation, then they commute.
(iii) Suppose now that two real matrices and commute and that has distinct eigenvalues. Show that for any eigenvector of the vector is a scalar multiple of . Deduce that there exists a common basis transformation that diagonalises both matrices.
(iv) Show that and satisfy the conditions in (iii) and find a matrix such that both of the matrices and are diagonal.
Paper 1, Section I, 1B
(a) Let
(i) Compute .
(ii) Find all complex numbers such that .
(b) Find all the solutions of the equation
(c) Let