CHAPTER 8�Supplementary Materials

8.1 Jordan form

8.2 Cayley-Hamilton theorem and its applications

8.3 More on the real and symmetric matrices

8.4 Singular value decomposition (SVD)

8.5 Final topics

中興大學 電機系 許舜斌 教授

Version: 101525

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8.1 Jordan form

Q: Why is a square matrix not diagonalizable?

A: Because the matrix has at least one eigenvalue whose

algebraic multiplicity is more than its dimension of eigenspace.

Two concepts:

• three parameters of an eigenvalue
• Jordan blocks

will be introduced before we present the Jordan form of a matrix

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What is a Jordan block?

What are the three parameters of an eigenvalue?

8.1 Jordan form

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The (AM,GM,index) for each eigenvalue in each matrix is:

8.1 Jordan form

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Q: Why is the Jordan form useful in dealing with the

non-diagonalizable matrices?

A: The Jordan form generalizes the concepts of diagonal matrix

and eigenvectors. Using this form, each eigenvalue in each

matrix has sufficient independent generalized eigenvectors.

• A diagonal form turns out to be a special Jordan form!!!
• A matrix is diagonalizable if and only if for each eigenvalue of the matrix, the index is 1, or equivalently, its AM equals GM.
• If all eigenvalues of a matrix are distinct, the matrix must be diagonalizable.

8.1 Jordan form

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Consequently,

8.1 Jordan form

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For example,

8.1 Jordan form

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The special structure of Jordan form allows one to show a very useful result known as the Cayley-Hamilton theorem, as detailed in the following section.

8.1 Jordan form

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• M is a called a nilpotent matrix if there exists a positive integer p such that M^p = 0.
• The smallest such p is called the index or degree of this nilpotent matrix. What makes the Jordan form so special can be seen from the following example:

8.2 Cayley-Hamilton theorem

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8.2 Cayley-Hamilton theorem

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8.2 Cayley-Hamilton theorem

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Ex.1

Sol.

However, this way needs to calculate the Jordan form, eigenvectors, and the inverse of a matrix. As we will see, the theory due to Cayley-Hamilton provides a simpler approach.

8.2 Cayley-Hamilton theorem

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Ex.1

8.2 Cayley-Hamilton theorem

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Ex.2

Sol.

8.2 Cayley-Hamilton theorem

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8.2 Cayley-Hamilton theorem

Ex.3

proof.

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8.3 More on the real and symmetric matrices

For example,

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Fundamental theorem of Hermitian matrices

proof. of p.1

8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

proof. of p.2

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8.3 More on the real and symmetric matrices

proof. of p.3

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8.3 More on the real and symmetric matrices

Quadratic form

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8.3 More on the real and symmetric matrices

Positive/Negative (semi)-definite matrices

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real

p.s.d.

symmetric

p.d.

classification of square matrices

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8.3 More on the real and symmetric matrices

Positive/Negative (semi)-definite matrices

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8.3 More on the real and symmetric matrices

Positive/Negative (semi)-definite matrices

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8.3 More on the real and symmetric matrices

Positive/Negative (semi)-definite matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

Real part

Imaginary part

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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If A and B are both square matrices of the same size, and A is nonsingular,

That means AB is similar to BA. Therefore, AB and BA have the same characteristic polynomial and thus the same set of eigenvalues. What happens if A and B have different sizes?

Cauchy’s interlace theorem has another version. Before mentioning that, we discuss a very important result on the eigenvalues of AB and BA.

8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.3 More on the real and symmetric matrices

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8.4 Singular value decomposition (SVD)

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8.4 Singular value decomposition (SVD)

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8.4 Singular value decomposition (SVD)

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8.4 Singular value decomposition (SVD)

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8.4 Singular value decomposition (SVD)

Eigenvectors of AA^T and A^TA

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8.4 Singular value decomposition (SVD)

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8.5 Final topics (I)

Relation between eigenvalues and minors

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8.5 Final topics (I)

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8.5 Final topics (I)

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8.5 Final topics (I) Cauchy-Binet formula

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8.5 Final topics (II) Gershgorin’s disk theorem

Bounds of eigenvalues

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8.5 Final topics (II)

Bounds of eigenvalues

The four Gershgorin disks centered at

(-3,0), (6,0), (5,0) and (1,0),

with radii 3, 11, 7 and 8 respectively, are drawn in the right figure. All the four eigenvalues of A are located

in the union: U1, of these 4 disks.

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8.5 Final topics (II)

Bounds of eigenvalues

Note that A^T and A have the same eigenvalues. We can consider the columns of A to obtain another four Gershgorin disks, centered at

(-3,0), (6,0), (5,0) and (1,0),

with radii 8, 6, 8 and 7 respectively, as shown in the right figure. All the four eigenvalues of A are located

in the union: U2, of these 4 disks as well.

In conclusion, we can take the intersection of U1 and U2 to obtain a tighter bound for the eigenvalues.

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8.5 More on Gershgorin’s theorem: I. disjoint case

eigenvalues

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8.5 More on Gershgorin’s theorem: II. Taussky’s theorem

Which one is nonsingular?

Which one has a convergent infinite power?

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8.5 More on Gershgorin’s theorem: II. Taussky’s theorem

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8.5 Final topics (III) Fibonacci’s sequence

Solving linear difference equation

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8.5 Final topics (IV)

Summary: relation between rank, nullity, singular value, eigenvalue?

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8.5 Final topics (v)

Solutions of Ax=b, A has the size mxn and rank r