CS6015: Linear Algebra and Random Processes

Lecture 19:  Algebraic and Geometric Multiplicity, Schur's theorem, Spectral theorem for Symmetric matrices, Trace of a matrix

Learning Objectives

What is algebraic and geometric multiplicity of an eigenvalue?

When would the two be equal?

What does Schur's theorem say about factorisation of a matrix?

What is Spectral theorem for symmetric matrices?

What is the trace of a matrix and how is it related to eigenvalues?

The Eigenstory

real

imaginary

distinct

repeating

\(A^\top\)

\(A^{-1}\)

\(AB\)

\(A^\top A\)

(basis)

powers of A

steady state

PCA

optimisation 

diagonalisation

\(A+B\)

\(U\)

\(R\)

\(A^2\)

\(A + kI\)

How to compute eigenvalues?

What are the possible values?

What are the eigenvalues of some special matrices ?

What is the relation between the eigenvalues of related matrices?

What do eigen values reveal about a matrix?

What are some applications in which eigenvalues play an important role?

Identity

Projection

Reflection

Markov

Rotation

Singular

Orthogonal

Rank one

Symmetric

Permutation

det(A - \lambda I) = 0

trace

determinant

invertibility 

rank

nullspace

columnspace

(Markov matrices)

(positive semidefinite matrices)

positive pivots

(independent eigenvectors)

(orthogonal eigenvectors)

... ...

(symmetric)

(where are we?)

(characteristic equation)
(desirable)

HW5

distinct values

independent eigenvectors

\(\implies\)

The bigger picture

* more than 1 value can repeat - e.g. in a projection matrix both the eigenvalues 1 and 0 may repeat

(questions)

What do we mean by saying that a matrix is diagonalizable?

Why do we care about diagonalizability?

What is the condition under which we will have \(n\) independent vectors even with repeating eigenvalues?

Multiplicity

For a given eigenvalue \(\lambda\)

Geometric Multiplicity(GM) is the number of independent eigenvectors for \(\lambda\)

Algebraic Multiplicity(AM) is the number of repetitions of \(\lambda\) among the eigenvalues

GM\leq AM

The good case (no repeating eigenvalues)

GM = AM=1~~\forall \lambda

Multiplicity

When would the number of independent eigenvectors be the same as the number of repetitions of \(\lambda\)?

Hint: How do we find the eigenvectors for a given \(\lambda\) ?

By solving \((A - \lambda I)\mathbf{x} = \mathbf{0}\)

\(\therefore\) number of independent eigenvectors = dimension of nullspace

\(\therefore\) GM = dimension of null space of \(A - \lambda I\)

If an eigenvalue repeats \(p\) times and the dimension of the nullspace of \((A-\lambda I)\) is also \(p\) then GM = AM

(i.e. when would GM be equal to AM)
(i.e. by finding vectors in the nullspace of 
(i.e.~~if~~rank(A - \lambda I) = n-p)
A - \lambda I)

The bigger picture

* more than 1 value can repeat - e.g. in a projection matrix both the eigenvalues 1 and 0 may repeat

Are there matrices which will always have \(n\) independent eigenvectors (even when one or more eigenvalues are repeating) ?

Yes, symmetric matrices

Real Symmetric matrices

we like real numbers as opposed to imaginary numbers

Properties

\(n\) real eigenvalues

\(n\) orthogonal eigenvectors

*

*

always diagonalisable 

*

the best possible basis
S = Q\Lambda Q^{-1} = Q\Lambda Q^{T}
even when there are repeating eigenvalues
(we will prove this soon)
(we will prove this soon)
(all elements are real and the matrix is symmetric)
(we will not prove this completely - not even in HW5 :-))

Real Symmetric matrices

(all elements are real and the matrix is symmetric)

Proof: 

Let \(\lambda = a + ib\) be an eigenvalue,

 \(\overline{\lambda} = a - ib\)

S\mathbf{x} = \lambda \mathbf{x}
\therefore S\overline{\mathbf{x}} = \overline{\lambda} \overline{\mathbf{x}}
(taking conjugate on both sides, S is real)
\therefore \overline{\mathbf{x}}^\top S = \overline{\mathbf{x}}^\top\overline{\lambda}
(taking transpose on both sides, S is symmetric)
multiply both sides by conjugate of x
multiply both sides by x
\overline{\mathbf{x}}^\top S\mathbf{x} = \overline{\mathbf{x}}^\top \lambda \mathbf{x}
\overline{\mathbf{x}}^\top S\mathbf{x} = \overline{\mathbf{x}}^\top \overline{\lambda} \mathbf{x}
\therefore \lambda \overline{\mathbf{x}}^\top \mathbf{x} = \overline{\lambda} \overline{\mathbf{x}}^\top \mathbf{x}
\therefore \lambda = \overline{\lambda}
\because \overline{\mathbf{x}}^\top \mathbf{x} \neq 0
\therefore a+ib = a-ib
\implies b = 0
Hence~proved!
Where did we use the property that the matrix is real symmetric?

Theorem: All the eigenvalues of a real symmetric matrix are real

Real Symmetric matrices

(all elements are real and the matrix is symmetric)

Theorem: Eigenvectors of a real symmetric matrix corresponding to different eigenvalues are orthogonal

Proof: 

Let~~S\mathbf{x} = \lambda_1 \mathbf{x}~~and~~S\mathbf{y} = \lambda_2 \mathbf{y}~~and~~\lambda_1 \neq \lambda_2
\mathbf{x}^\top\lambda_1\mathbf{y}
= (\lambda_1\mathbf{x})^\top\mathbf{y}
= (S\mathbf{x})^\top\mathbf{y}
= \mathbf{x}^\top S^\top\mathbf{y}
= \mathbf{x}^\top S\mathbf{y}
= \mathbf{x}^\top \lambda_2\mathbf{y}
\therefore \lambda_1\mathbf{x}^\top\mathbf{y}=\lambda_2\mathbf{x}^\top\mathbf{y}
\implies \lambda_1=\lambda_2~~or~~\mathbf{x}^\top\mathbf{y}=0
But~~\lambda_1\neq\lambda_2
\therefore~~\mathbf{x}^\top\mathbf{y}=0
(Hence proved)

Real Symmetric matrices

Proved

Properties

\(n\) real eigenvalues

What about the case when some eigenvalues repeat?

*

*

always diagonalisable 

*

Proved for the case when eigenvalues are not repeating
(all elements are real and the matrix is symmetric)
Follows from Theorem 2 when the eigenvalues are not repeating

\(n\) orthogonal eigenvectors

(see next few slides)
S = Q\Lambda Q^{-1} = Q\Lambda Q^{T}

Schur's Theorem

Theorem : If \(A\) is a square matrix with real eigenvalues, then there is an orthogonal matrix \(Q\) and an upper triangular matrix \(T\) such that,

\(A = QTQ^\top\)

Corollary : If \(A\) is a square symmetric matrix with real eigenvalues, then there is an orthogonal matrix \(Q\) and a diagonal matrix \(T\) such that,

\(A = QTQ^\top\)

Proof:

A = QTQ^\top
A^\top = (Q T Q^\top)^\top = Q T^\top Q^\top
But~~A = A^\top
\therefore QTQ^\top = QT^\top Q^\top
\therefore T = T^\top
upper triang.
lower triang.
Hence, T must be diagonal
(we will not prove this)
from Schur's Theorem
symmetric

Spectral Theorem

Theorem : If \(A\) is a real square symmetric matrix , then

every eigenvalue of \(A\) is a real number

eigenvectors corresponding to different eigenvalues are orthogonal

\(A\) can be diagonalised as \(Q D Q^\top\) 

(orthogonal matrix of eigenvectors of A)
(diagonal matrix of eigenvalues of A)

*

*

*

Proved
Proved 
therefore q_i's must be eigenvectors and d_i's must be eigenvalues
From~~corollary~~of~~Schur's~~theorem

Proof (of 3rd part):

A = QTQ^\top
\therefore AQ = QT
T is diagonal
\therefore A \begin{bmatrix} \uparrow&\uparrow&\uparrow \\ q_1&\dots&q_n \\ \downarrow&\downarrow&\downarrow \\ \end{bmatrix} = \begin{bmatrix} \uparrow&\uparrow&\uparrow \\ q_1&\dots&q_n \\ \downarrow&\downarrow&\downarrow \\ \end{bmatrix} \begin{bmatrix} d_1&\dots&0 \\ 0&\dots&0 \\ 0&\dots&d_n \\ \end{bmatrix}
\therefore \begin{bmatrix} \uparrow&\uparrow&\uparrow \\ Aq_1&\dots&Aq_n \\ \downarrow&\downarrow&\downarrow \\ \end{bmatrix} = \begin{bmatrix} \uparrow&\uparrow&\uparrow \\ d_1q_1&\dots&d_nq_n \\ \downarrow&\downarrow&\downarrow \\ \end{bmatrix}
(even if there are repeating eigenvalues)
even if there are repeating eigenvalues

Spectral Theorem

(informal conclusion) 

Real symmetric square matrices are the best possible matrices

*

always diagonalisable

*

with real eigenvalues

*

with orthogonal eigenbasis

we love diagonalizability
we love real numbers
we love orthogonal basis

One last property about symmetric matrices

sorry for squeezing it in here

number of positive pivots = number of positive eigenvalues

thus connecting the two halves of the course

(HW5)

The Eigenstory

real

imaginary

distinct

repeating

\(A^\top\)

\(A^{-1}\)

\(AB\)

\(A^\top A\)

(basis)

powers of A

steady state

PCA

optimisation 

diagonalisation

\(A+B\)

\(U\)

\(R\)

\(A^2\)

\(A + kI\)

How to compute eigenvalues?

What are the possible values?

What are the eigenvalues of some special matrices ?

What is the relation between the eigenvalues of related matrices?

What do eigen values reveal about a matrix?

What are some applications in which eigenvalues play an important role?

Identity

Projection

Reflection

Markov

Rotation

Singular

Orthogonal

Rank one

Symmetric

Permutation

det(A - \lambda I) = 0

trace

determinant

invertibility 

rank

nullspace

columnspace

(Markov matrices)

(positive semidefinite matrices)

positive pivots

(independent eigenvectors)

(orthogonal eigenvectors)

... ...

(symmetric)

(where are we?)

(characteristic equation)
(desirable)

HW5

distinct values

independent eigenvectors

\(\implies\)

Two properties

The trace of a matrix is the sum of the diagonal elements of the matrix

tr(A) = \sum_{i=1}^n a_{ii}
defined only for square matrices

If \(\lambda_1, \lambda_2, \dots, \lambda_n\) are the eigenvalues of a matrix \(A\) then

tr(A) = \sum_{i=1}^n \lambda_{i}
det(A) = \prod_{i=1}^n \lambda_{i}

(Proof in HW5)

(Proof in HW5)

The Eigenstory

real

imaginary

distinct

repeating

\(A^\top\)

\(A^{-1}\)

\(AB\)

\(A^\top A\)

(basis)

powers of A

steady state

PCA

optimisation 

diagonalisation

\(A+B\)

\(U\)

\(R\)

\(A^2\)

\(A + kI\)

How to compute eigenvalues?

What are the possible values?

What are the eigenvalues of some special matrices ?

What is the relation between the eigenvalues of related matrices?

What do eigen values reveal about a matrix?

What are some applications in which eigenvalues play an important role?

Identity

Projection

Reflection

Markov

Rotation

Singular

Orthogonal

Rank one

Symmetric

Permutation

det(A - \lambda I) = 0

trace

determinant

invertibility 

rank

nullspace

columnspace

(Markov matrices)

(positive definite matrices)

positive pivots

(independent eigenvectors)

(orthogonal eigenvectors)

... ...

(symmetric)

(where are we?)

(characteristic equation)
(desirable)

HW5

distinct values

independent eigenvectors

\(\implies\)

Positive Definite Matrices

pivots, eigenvalues, determinants, \( x^\top Ax\) > 0

write quadratic equation for 2 6 2 7

geometry by replacing the 7 by 18, 20

Learning Objectives

What is algebraic and geometric multiplicity of an eigenvalue?

When would the two be equal?

What does Schur's theorem say about factorisation of a matrix?

What is Spectral theorem for symmetric matrices?

What is the trace of a matrix and how is it related to eigenvalues?