Linear least squares (mathematics)

In linear algebra, a Householder transformation (also known as Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm. The Householder transformation was introduced in 1958 by Alston Scott Householder.

Its analogue over general inner product spaces is the Householder operator.

Definition and properties

The reflection hyperplane can be defined by a unit vector v (a vector with length 1) which is orthogonal to the hyperplane. The reflection of a point x about this hyperplane is:

$x-2⟨v,x⟩v=x-2v(v^{\text{H}}x),$

where v is given as a column unit vector with Hermitian transpose v^H. This is a linear transformation given by the Householder matrix:

$P=I-2v{v}^{\text{H}}$, where I is the identity matrix.

The Householder matrix has the following properties:

• it is Hermitian: $P=P^{\text{H}},$
• it is unitary: $P^{-1}=P^{\text{H}},$
• hence it is involutary: $P^2=I$.
• A Householder matrix has eigenvalues $\pm 1$. To see this, notice that if $u$ is orthogonal to the vector $v$ which was used to create the reflector, then $Pu=u$, i.e., 1 is an eigenvalue of multiplicity $n-1$, since there are $n-1$ independent vectors orthogonal to $v$. Also, notice $Pv=-v$, and so -1 is an eigenvalue with multiplicity 1.
• The determinant of a Householder reflector is -1, since the determinant of a matrix is the product of its eigenvalues.

Applications

In geometric optics, specular reflection can be expressed in terms of the Householder matrix.

Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the (i, i) minors of that product.

They are also widely used for tridiagonalization of symmetric matrices and for transforming non-symmetric matrices to a Hessenberg form.

Tridiagonalization

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. This procedure is taken from the book: Numerical Analysis, Burden and Faires, 8th Edition. In the first step, to form the Householder matrix in each step we need to determine $\alpha$ and r, which are:

$\alpha =-\mathrm{sgn}(a_{21})\sqrt{\sum _{j=2}^n{a}_{j1}^2}$;

$r=\sqrt{\frac{1}{2}({\alpha }^2-a_{21}\alpha )}$;

From $\alpha$ and r, construct vector v:

$v^{(1)}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right],$

where $v_1=0;$, $v_2=\frac{a_{21}-\alpha }{2r}$, and

$v_k=\frac{a_{k1}}{2r}$ for each k=3,4 ..n

Then compute:

$P^1=I-2v^{(1)}(v^{(1)}{)}^{\text{T}}$

$A^{(2)}=P^{1}A{P}^1$

Having found $P^1$ and computed $A^{(2)}$ the process is repeated for k =2, 3, ..., n-1 as follows:

$\alpha =-\mathrm{sgn}(a_{k+1,k}^k)\sqrt{\sum _{j=k+1}^n(a_{jk}^k{)}^2}$;

$r=\sqrt{\frac{1}{2}({\alpha }^2-a_{k+1,k}^k\alpha )}$;

$v_1^k=v_2^k=\cdots =v_k^k=0;$

$v_{k+1}^k=\frac{a_{k+1,k}^k-\alpha }{2r}$

$v_j^k=\frac{a_{jk}^k}{2r}$ for j = k + 2; k + 3, ..., n

$P^k=I-2v^{(k)}(v^{(k)}{)}^{\text{T}}$

$A^{(k+1)}=P^k{A}^{(k)}{P}^k$

Continuing in this manner, the tridiagonal and symmetric matrix is formed.

Examples

This example is taken from the book "Numerical Analysis" by Richard L. Burden (Author), J. Douglas Faires. In this example, the given matrix is transformed to the similar tridiagonal matrix A₂ by using Householder Method.

$\mathbf{A}=\left[\begin{array}{cccc}4& 1& -2& 2\\ 1& 2& 0& 1\\ -2& 0& 3& -2\\ 2& 1& -2& -1\end{array}\right],$

Following those steps in Householder Method. We have:

The first Householder matrix:

Q₁ $=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1/3& 2/3& -2/3\\ 0& 2/3& 2/3& 1/3\\ 0& -2/3& 1/3& 2/3\end{array}\right],$

A₁ = Q₁AQ₁ = $\left[\begin{array}{cccc}4& -3& 0& 0\\ -3& 10/3& 1& 4/3\\ 0& 1& 5/3& -4/3\\ 0& 4/3& -4/3& -1\end{array}\right],$

Used A₁ to form Q₂ =$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -3/5& -4/5\\ 0& 0& -4/5& 3/5\end{array}\right],$

A₂ = Q₂A₁Q₂=$\left[\begin{array}{cccc}4& -3& 0& 0\\ -3& 10/3& -5/3& 0\\ 0& -5/3& -33/25& 68/75\\ 0& 0& 68/75& 149/75\end{array}\right],$

As we can see, the final result is a tridiagonal symmetric matrix which is similar to the original one. The process finished after 2 steps.

Computational and Theoretical Relationship to other Unitary Transformations

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$\frac{\text{Trace}(U{U}^{\text{H}})}{N}=\frac{\sum _{j=2}^N|{\lambda }_j{|}^2}{N}=1,\text{det}(U{U}^{\text{H}})=\prod _{j=1}^N|{\lambda }_j{|}^2=1.$

Since arithmetic and geometric means are equal iff the variables are constant, see, inequality of arithmetic and geometric means, we establish the claim of unit modulus.

For the case of real valued unitary matrixes we obtain orthogonal matrices, $U{U}^{\text{T}}=I.$ In this case all eigenvalues are real, and so the unit modulus eigenvalue constraint is replaced by the binary constraint that all eigenvalues lie in the set {+1,-1}. It follows rather readily (see orthogonal matrix) that any orthogonal matrix can be decomposed into a product of 2 by 2 rotations, called Givens Rotations, and Householder reflections. This is appealing intuitively since multiplication of a vector by an orthogonal matrix preserves the length of that vector, and rotations and reflections exhaust the set of (real valued) geometric operations that render invariant a vector's length.

The Householder transformation was shown to have a one to one relationship with the canonical coset decomposition of unitary matrices defined in group theory, which can be used to parametrize unitary operators in a very efficient manner.

Finally we note that a single Householder Transform, unlike a solitary Givens Transform, can act on all columns of a matrix, and as such exhibits the lowest computational cost for QR decomposition and Tridiagonalization. The penalty for this "computational optimality" is, of course, that Householder operations cannot be as deeply or efficiently parallelized. As such Householder is preferred for dense matrices on sequential machines, whilst Givens is preferred on sparse matrices, and/or parallel machines.

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang (Herein Householder Transformation is cited as a top 10 algorithm of this century)

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External links

• Householder Transformations