MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

MAA704: Matrix factorization and canonicalforms

Karl Lundengard

November 23, 2012

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Contents of todays lecture

I Some interesting / useful / important properties ofmatrices

I Matrix factorization

I Canonical forms

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Matrix factorization

I Rewriting a matrix as a product of several matrices.

I Choosing these factor matrices wisely can make problemseasier to solve.

I Also known as matrix decomposition

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Diagonalizable matrix

DefinitionIf B = S−1DS where D is a diagonal matrix then B is diagonal.

Motivation.Using elementary row operations we want to turn Bx = y intoDx = y . This can be written as SBx = Sy . Since elementaryrow operations are invertible SBS−1Sx = Sy . Let x = Sx andy = Sy , then

D = SBS−1 ⇔ B = S−1DS

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Triangular matrix

A =

F F . . . F0 F . . . F...

.... . .

...0 0 . . . F

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Triangular matrix

I Can be lower (left) or upper (right) triangular

I Easy to solve equation systems involving triangularmatrices

I Diagonal values are also eigenvalues

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Hessenberg matrix

A =

F F F · · · F F FF F F · · · F F F0 F F · · · F F F0 0 F · · · F F F...

......

. . ....

......

0 0 0 · · · F F F0 0 0 · · · 0 F F

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Hessenberg matrix

I ’Almost’ triangular

I Multiplication of two Hessenberg matrices gives a newHessenberg matrix

I Useful in the QR-method (Lecture 9)

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Hermitian matrix

DefinitionThe Hermitian conjugate of a matrix A is denoted AH and isdefined by (AH)ij = (A)ji .

DefinitionA matrix is said to be Hermitian (or self-adjoint) if AH = A

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Hermitian matrix

I Notice the similarities with a symmetric matrix A> = A.

I All eigenvalues real.

I Always diagonalizable.

I Important in theoretical physics, quantum physics,electroengineering and in certain problems in statistics.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Unitary matrices

DefinitionA matrix, A, is said to be unitary if AH = A−1.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Properties of unitary matrices

TheoremLet U be a unitary matrix, then

a) U is always invertible.

b) U−1 is also unitary.

c) | det(U)| = 1

d) (UV )H = (UV )−1 if V is also unitary.

e) For any λ that is an eigenvalue of U, λ = e iω, 0 ≤ ω ≤ 2π.

f) Let v be a vector, then |Uv | = |v | (for any vector norm1).

g) The rows/columns of U are orthonormal, that is Ui .UHj . = 0,

i 6= j , Uk.UHk. = 1.

h) U preserves eigenvalues.

1see lecture 7

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Example of a unitary matrix

I The C matrix below rotates a vector by the angle θaround the x-axis

C =

1 0 00 cos(θ) − sin(θ)0 sin(θ) cos(θ)

and is a unitary matrix.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Positive definite matrix

DefinitionWe consider a square symmetric real valued n × n matrix A,then:

I A is positive definite if xTAx is positive for all non-zerovectors x .

I A is positive semidefinite if xTAx is non-negative for allnon-zero vectors x .

I A is positive definite ⇔ λ > 0 for all λ eigenvalue of A.

I Can also define negative definite and semi-definitematrices, more about this in lecture 10.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Matrix factorization

I Diagonalizable A = S−1DS with D diagonalI Other important factorizations:

I Spectral factorization QΛQ−1

I LU-factorizationI Cholesky factorization GGH

I QR-factorizationI Rank factorization CFI Jordan canonical form S−1JSI Singular value factorization UΣV H

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Spectral factorization

I Spectral factorization is a special version of diagonalfactorization.

I It is sometimes referred to as eigendecomposition.

I Let A be an square (n × n) matrix with linearlyindependent rows. Then

A = QΛQ−1

where AQ.i = ΛiiQ.i for all 1 ≤ i ≤ n.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Rank factorization

I Let A be an m × n matrix with rank(A) = r (A has rindependent rows/columns). Then

A = CF

where C ∈Mm×r and F ∈Mr×n

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Rank factorization

I How can we find this factorization?

I Rewrite matrix on an echelon form

B =

0 ~ ∗ ∗ ∗ ∗ ∗ ∗0 0 0 ~ ∗ ∗ ∗ ∗0 0 0 0 ~ ∗ ∗ ∗0 0 0 0 0 ~ ∗ ∗0 0 0 0 0 0 0 ~0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Rank factorization

I Create C by removing all columns in A that correspondsto a non-pivot column in B.

I In this example

C =[A.2 A.4 A.5 A.6 A.8

]I Create F by removing all zero rows in B.

I In this example

F =[B1. B2. B3. B4. B5.

]

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

LU-factorization

I A = LR = LU, L lower (left) triangular

I L is a n × n lower triangular matrix.

I U is a n ×m upper triangular matrix.

I Solve Ax = L(Ux) = b by first solving Ly = b and thensolve Ux = y . Both these systems are easy to solve sinceL and U are both triangular.

I Not every matrix A have a LU factorization, not evenevery square invertible matrix.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

LUP-factorization

TheoremEvery n ×m matrix A have a matrix factorization

PA = LU

. where

I P is a n × n permutation matrix.

I L is a n × n lower triangular matrix.

I U is a n ×m upper triangular matrix.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Cholesky factorization

I Systems involving triangular matrices are often easy tosolve.

I Try to rewrite a matrix as a product that contains atriangular matrix seems like a good idea.

I One way is using LU-factorization where PA = LU whereP is a permutation matrix, L is a lower- and U is an uppertriangular matrix.

I More on the LU-factorization in lecture 10.

I There is also the Cholesky factorization, A = GGH , whereA is Hermitian and positive-definite and G is lowertriangular.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Cholesky factorization

I Consider the equation Ax = y . If a can be Choleskyfactorized, A = GGH , this equation can be turned into twonew equations: {

Gz = y

GHx = z

both of these equations are easy to solve.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Applications of Cholesky factorization

I Are there any interesting matrices that can be easyCholesky factorized?

I Any covariance matrix is positive-definite and anycovariance matrix based on measured data is going to besymmetric and real-valued. From the last two properties itfollows that this matrix is Hermitian.

I Example application: generating variates according to amultivariate distribution with covariance matrix Σ andexpected value µUsing the Cholesky factorization you get the simpleformula X = µ+ G>Z where X is the variate, Σ = GGH

and Z is a vector of standard normal variates.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

QR factorization

TheoremEvery n ×m matrix A have a matrix decomposition

A = QR

. where

I R is a n ×m upper triangular matrix..

I Q is a n × n unitary matrix.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

QR factorization

I Given a QR-factorization we can solve a linear systemAx = b by solving Rx = Q−1b = QHb. Which is can bedone fast since R is a triangular matrix.

I QR-factorization can also used in solving the linear leastsquare problem.

I It plays an important role in the QR-method used tocalculate eigenvalues of a matrix numerically.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Canonical form

I A canonical form is a standard way of describing an object.

I There can be several different kinds of canonical forms foran object.

I Some examples for matrices:I Diagonal form (for diagonalizable matrices)I Reduced row echelon form (for all matrices)I Jordan canonical form (for square matrices)I Singular value factorization form (for all matrices)

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Reduced row echelon form

DefinitionA matrix is written on reduced row echelon form when they arewritten on echelon form and their pivot elements are all equalto one and all other elements in a pivot column are zero.

B =

0 1 ∗ 0 0 0 ∗ 00 0 0 1 0 0 ∗ 00 0 0 0 1 0 ∗ 00 0 0 0 0 1 ∗ 00 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

TheoremAll matrices are similar to some reduced row echelon matrix.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Jordan normal form

Definition (Jordan block)

A Jordan block is a square matrix of the form

Jm(λ) =

λ 1 0 . . . 00 λ 1 . . . 0...

.... . .

. . ....

0 0 . . . λ 10 0 . . . 0 λ

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Jordan normal form

Definition (Jordan matrix)

A Jordan matrix is a square matrix of the form

J =

Jm1(λ1) 0 . . . 0

0 Jm2(λ2) . . . 0...

.... . .

...0 0 . . . Jm1(λk)

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Jordan normal form

TheoremAll square matrices are similar to a Jordan matrix. The Jordanmatrix is unique except for the order of the Jordan blocks. ThisJordan matrix is called the Jordan normal form of the matrix.

Theorem (Some other interesting properties of the Jordannormal form)

Let A = S−1JS

a) The eigenvalues of J is the same as the diagonal elementsof J.

b) J has one eigenvector per Jordan block.

c) The rank of J is equal to the number of Jordan blocks.

d) The normal form is sensitive to perturbations. This meansthat a small change in the normal form can mean a largechange in the A matrix and vice versa.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Singular value factorization

TheoremAll A ∈Mm×n can be factorized as

A = UΣV H

where U and V are unitary matrices and

Σ =

[Sr 00 0

]where Sr is a diagonal matrix with r = rank(A). The diagonalelements of Sr are called the singular values. The singularvalues are uniquely determined by the matrix A (but notnecessarily their order).

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Singular value factorization

I Very often referred to as the SVD (singular valuedecomposition).

I Used a lot in statistics and information processing.

I Can be used to quantify many different qualities ofmatrices, more on this in later lectures.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Similar matrices

I In everyday language two matrices are ’similar’ if theyhave almost the same elements or structure. But there isalso a precise mathematical relation between two matricesthat is called similar.

DefinitionTwo matrices, A and B, are similar if A = S−1BS .

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Interesting properties of similar matrices

I Similar matrices share several properties:I Eigenvalues (but generally not eigenvectors)I DeterminantI TraceI Rank

I We have already seen some examples of why similarmatrices are interesting:

I Diagonalizable matrices A = S−1BSI Permutation matrices A = PBP>

I Jordan normal form A = S−1JS

I Similarity between matrices mean they represent the samelinear mapping described in different basis. More on this inlecture 8.

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Summary

I Triangular and Hessenberg matrices

I Hermitian matrices

I Unitary matrices

I Positive definite matrices

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Summary

I Matrix factorizationI Spectral factorization QΛQ−1

I LU-factorizationI Cholesky factorization GGH

I QR-factorizationI Rank factorization CFI Jordan canonical form S−1JSI Singular value factorization UΣV H

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Next lecture

I Matrix functions and matrix equations with SergeiSilvestrov

MAA704:Matrix

factorizationand canonical

forms

Matrixfactorization

Matrixproperties

Triangularmatrix

Hessenbergmatrix

Hermitian matrix

Unitary matrices

Positive definitematrix

Matrixfactorization

Spectralfactorization

Rankfactorization

LU factorization

Choleskyfactorization

QR factorization

Canonicalforms

Reduced rowechelon form

Jordan normalform

Singular valuefactorization

Similar matrices

Summary

Remember the seminar assignment!

Remember the projects!

Have a nice weekend!