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In mathematics, a '''Hermitian matrix''' (or '''self-adjoint matrix''') is a [[square matrix]] with [[complex number|complex]] entries that is equal to its own [[conjugate transpose]]—that is, the element in the {{mvar|i}}-th row and {{mvar|j}}-th column is equal to the [[complex conjugate]] of the element in the {{mvar|j}}-th row and {{mvar|i}}-th column, for all indices {{mvar|i}} and {{mvar|j}}:
 
:<math>a_{ij} = \overline{a_{ji}}</math> or <math>A^T = \overline A</math>, in matrix form.
 
Hermitian matrices can be understood as the complex extension of real [[symmetric matrix|symmetric matrices]].
 
If the conjugate transpose of a matrix <math>A</math> is denoted by <math>A^\dagger</math>, then the Hermitian property can be written concisely as
 
:<math> A = A^\dagger\,.</math>
 
Note that asterisk <math>A^*</math> is commonly used to represent <math>A^\dagger</math> although some authors have been known to use it to represent <math>\overline A</math> which can create confusion.
 
Hermitian matrices are named after [[Charles Hermite]], who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having [[Eigenvalues and eigenvectors|eigenvalues]] always real.
 
== Examples ==
See the following example:
 
:<math>
\begin{bmatrix}
2 & 2+i & 4 \\
2-i & 3 & i \\
4 & -i & 1 \\
\end{bmatrix}
</math>
 
The diagonal elements must be [[real number|real]], as they must be their own complex conjugate.
 
Well-known families of [[Pauli matrices]], [[Gell-Mann matrices]] and their generalizations are Hermitian. In [[theoretical physics]] such Hermitian matrices are often multiplied by [[imaginary number|imaginary]] coefficients,<ref>
{{cite book |title=The geometry of physics: an introduction |last=Frankel |first=Theodore |authorlink=Theodore Frankel |year=2004 |publisher=[[Cambridge University Press]] |isbn=0-521-53927-7 |page=652 |url=http://books.google.ru/books?id=DUnjs6nEn8wC&lpg=PA652&dq=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&pg=PA652#v=onepage&q&f=false }}
</ref><ref>
[http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf Physics 125 Course Notes] at [[California Institute of Technology]]
</ref> which results in ''skew-Hermitian'' matrices (see [[#facts|below]]).
 
== Properties ==
 
* The entries on the [[main diagonal]] (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate. A matrix that has only real entries is Hermitian [[if and only if]] it is a [[symmetric matrix]], i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.
 
* Every Hermitian matrix is a [[normal matrix]].
 
* The finite-dimensional [[spectral theorem]] says that any Hermitian matrix can be [[diagonalizable matrix|diagonalized]] by a [[unitary matrix]], and that the resulting diagonal matrix has only real entries. This implies that all [[eigenvectors|eigenvalue]]s of a Hermitian matrix {{mvar|A}} are real, and that {{mvar|A}} has {{mvar|n}} linearly independent [[eigenvector]]s. Moreover, it is possible to find an [[orthonormal basis]] of {{math|'''C'''<sup>''n''</sup>}} consisting of {{mvar|n}} eigenvectors of {{mvar|A}}.
 
* The sum of any two Hermitian matrices is Hermitian, and the [[inverse matrix|inverse]] of an invertible Hermitian matrix is Hermitian as well. However, the [[matrix multiplication|product]] of two Hermitian matrices {{mvar|A}} and {{mvar|B}} is Hermitian if and only if {{math|1=''AB'' = ''BA''}}. Thus {{math|''A''<sup>''n''</sup>}} is Hermitian if {{mvar|A}} is Hermitian and {{mvar|n}} is an integer.
 
* The Hermitian complex {{mvar|n}}-by-{{mvar|n}} matrices do not form a [[vector space]] over the [[complex number]]s, since the identity matrix {{math|''I''<sub>n</sub>}} is Hermitian, but {{math|''i'' ''I''<sub>n</sub>}} is not. However the complex Hermitian matrices ''do'' form a vector space over the [[real numbers]] {{math|'''R'''}}.  In the {{math|2''n''<sup>2</sup>}}-[[dimension of a vector space|dimensional]] vector space of complex {{math|''n'' × ''n''}} matrices over {{math|'''R'''}}, the complex Hermitian matrices form a subspace of dimension {{math|''n''<sup>2</sup>}}. If {{math|''E''<sub>''jk''</sub>}} denotes the {{mvar|n}}-by-{{mvar|n}} matrix with a 1 in the {{math|''j'',''k''}} position and zeros elsewhere, a basis can be described as follows:
::<math>\; E_{jj}</math> for <math>1\leq j\leq n</math> ({{mvar|n}} matrices)
:together with the set of matrices of the form
::<math>\; E_{jk}+E_{kj}</math> for <math>1\leq j<k\leq n </math>    ({{sfrac|''n''<sup>2</sup> − ''n''|2}} matrices)
:and the matrices
::<math>\; i(E_{jk}-E_{kj})</math> for <math>1\leq j<k\leq n </math>    ({{sfrac|''n''<sup>2</sup> − ''n''|2}} matrices)
:where <math>i</math> denotes the complex number <math>\sqrt{-1}</math>, known as the [[imaginary unit]].
 
* If {{mvar|n}} orthonormal eigenvectors <math>u_1,\dots,u_n</math> of a Hermitian matrix are chosen and written as the columns of the matrix {{mvar|U}}, then one [[Eigendecomposition of a matrix|eigendecomposition]] of {{mvar|A}} is <math> A = U \Lambda U^\dagger</math> where <math> U U^\dagger = I=U^\dagger U</math> and therefore
::<math> A = \sum _j \lambda_j u_j u_j ^\dagger </math>,
:where <math>\lambda_j</math> are the eigenvalues on the diagonal of the diagonal matrix <math>\; \Lambda </math>.
 
== Further properties ==
{{anchor|facts}}Additional facts related to Hermitian matrices include:
* The sum of a square matrix and its conjugate transpose <math>(C + C^{\dagger})</math> is Hermitian.
* The difference of a square matrix and its conjugate transpose <math>(C - C^{\dagger})</math> is [[skew-Hermitian matrix|skew-Hermitian]] (also called antihermitian). This implies that [[commutator]] of two Hermitian matrices is skew-Hermitian.
* An arbitrary square matrix {{mvar|C}} can be written as the sum of a Hermitian matrix {{mvar|A}} and a skew-Hermitian matrix {{mvar|B}}:
::<math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^{\dagger}) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^{\dagger}).</math>
* The determinant of a Hermitian matrix is real:
:: Proof: <math> \det(A) = \det(A^\mathrm{T})\quad \Rightarrow \quad \det(A^\dagger) = \det(A)^* </math>
:: Therefore if <math>A=A^\dagger\quad \Rightarrow \quad \det(A) = \det(A)^*.</math>
:(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
 
==See also==
*[[Skew-Hermitian matrix]] (anti-Hermitian matrix)
*[[Haynsworth inertia additivity formula]]
*[[Hermitian form]]
*[[Self-adjoint operator]]
*[[Unitary matrix]]
 
==References==
{{reflist}}
 
==External links==
*{{springer|title=Hermitian matrix|id=p/h047070}}
*[http://people.ofset.org/~ckhung/b/la/hermitian.en.php Visualizing Hermitian Matrix as An Ellipse with Dr. Geo], by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
*{{MathPages|id=home/kmath306/kmath306|title=Hermitian Matrices}}
 
[[Category:Matrices]]

Revision as of 17:25, 31 October 2013

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose—that is, the element in the Template:Mvar-th row and Template:Mvar-th column is equal to the complex conjugate of the element in the Template:Mvar-th row and Template:Mvar-th column, for all indices Template:Mvar and Template:Mvar:

aij=aji‾ or AT=A‾, in matrix form.

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

If the conjugate transpose of a matrix A is denoted by A†, then the Hermitian property can be written concisely as

A=A†.

Note that asterisk A* is commonly used to represent A† although some authors have been known to use it to represent A‾ which can create confusion.

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real.

Examples

See the following example:

[22+i42−i3i4−i1]

The diagonal elements must be real, as they must be their own complex conjugate.

Well-known families of Pauli matrices, Gell-Mann matrices and their generalizations are Hermitian. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices (see below).

Properties

  • The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.
  • The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix Template:Mvar are real, and that Template:Mvar has Template:Mvar linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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  • The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices Template:Mvar and Template:Mvar is Hermitian if and only if Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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    This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

    A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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    There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

    This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

    A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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    Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. matrices over Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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    There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

    This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

    A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

    If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

    Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., the complex Hermitian matrices form a subspace of dimension Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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    There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

    This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

    A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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    There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

    This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

    A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

    If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

    Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. denotes the Template:Mvar-by-Template:Mvar matrix with a 1 in the Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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    There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

    This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

    A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

    If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

    Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. position and zeros elsewhere, a basis can be described as follows:
Ejj for 1≤j≤n (Template:Mvar matrices)
together with the set of matrices of the form
Ejk+Ekj for 1≤j<k≤n (Electrical Engineer Stephan from Merrickville-Wolford, likes to spend time body building, property developers ec in singapore singapore and storytelling. Gets inspiration through travel and just spent 6 days at Würzburg Residence with the Court Gardens. matrices)
and the matrices
i(Ejk−Ekj) for 1≤j<k≤n (Electrical Engineer Stephan from Merrickville-Wolford, likes to spend time body building, property developers ec in singapore singapore and storytelling. Gets inspiration through travel and just spent 6 days at Würzburg Residence with the Court Gardens. matrices)
where i denotes the complex number −1, known as the imaginary unit.
A=∑jλjujuj†,
where λj are the eigenvalues on the diagonal of the diagonal matrix Λ.

Further properties

<facts>...</facts>Additional facts related to Hermitian matrices include:

  • The sum of a square matrix and its conjugate transpose (C+C†) is Hermitian.
  • The difference of a square matrix and its conjugate transpose (C−C†) is skew-Hermitian (also called antihermitian). This implies that commutator of two Hermitian matrices is skew-Hermitian.
  • An arbitrary square matrix Template:Mvar can be written as the sum of a Hermitian matrix Template:Mvar and a skew-Hermitian matrix Template:Mvar:
C=A+BwithA=12(C+C†)andB=12(C−C†).
  • The determinant of a Hermitian matrix is real:
Proof: det⁡(A)=det⁡(AT)⇒det⁡(A†)=det⁡(A)*
Therefore if A=A†⇒det⁡(A)=det⁡(A)*.
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  2. ↑ Physics 125 Course Notes at California Institute of Technology
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