Merkle–Hellman knapsack cryptosystem: Difference between revisions

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In [[linear algebra]], the '''identity matrix''' or '''unit matrix''' of size ''n'' is the ''n'' &times; ''n'' [[square matrix]] with ones on the [[main diagonal]] and zeros elsewhere. It is denoted by ''I''<sub>''n''</sub>, or simply by ''I'' if the size is immaterial or can be trivially determined by the context. (In some fields, such as [[quantum mechanics]], the identity matrix is denoted by a boldface one, '''1'''; otherwise it is identical to ''I''.) Some mathematics books use ''U'' and ''E'' to represent the Identity Matrix (meaning "Unit Matrix" and the German word "Einheitsmatrix",<ref>[http://mathworld.wolfram.com/IdentityMatrix.html "Identity Matrix"] on [[MathWorld]];</ref> respectively), although ''I'' is considered more universal.{{fact|date=November 2013}}
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:<math>
I_1 = \begin{bmatrix}
1 \end{bmatrix}
,\
I_2 = \begin{bmatrix}
1 & 0 \\
0 & 1 \end{bmatrix}
,\
I_3 = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix}
,\ \cdots ,\
I_n = \begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1 \end{bmatrix}
</math>
 
When ''A'' is ''m''×''n'', it is a property of [[matrix multiplication]] that
:<math>I_mA = AI_n = A. \,</math>
In particular, the identity matrix serves as the unit of the [[ring (mathematics)|ring]] of all ''n''×''n'' matrices, and as the [[identity element]] of the [[general linear group]] GL(''n'') consisting of all [[invertible matrix|invertible]] ''n''×''n'' matrices. (The identity matrix itself is invertible, being [[Involutory matrix|its own inverse]].)
 
Where ''n''×''n'' matrices are used to represent [[linear transformation]]s from an ''n''-dimensional vector space to itself, ''I<sub>n</sub>'' represents the [[identity function]], regardless of the [[Basis (linear algebra)|basis]].
 
The ''i''th column of an identity matrix is the [[unit vector]] ''e<sub>i</sub>''.  It follows that the [[determinant]] of the identity matrix is&nbsp;1 and the [[trace (linear algebra)|trace]] is&nbsp;''n''.
 
Using the notation that is sometimes used to concisely describe [[diagonal matrix|diagonal matrices]], we can write:
:<math> I_n = \mathrm{diag}(1,1,...,1). \,</math>
 
It can also be written using the [[Kronecker delta]] notation:
:<math>(I_n)_{ij} = \delta_{ij}. \,</math>
 
The identity matrix also has the property that, when it is the product of two square matrices, the matrices can be said to be the inverse of one another.
 
The identity matrix of a given size is the only [[idempotent matrix]] of that size having [[full rank]]. That is, it is the only matrix such that (a) when multiplied by itself the result is itself, and (b) all of its rows, and all of its columns, are [[linear independence|linearly independent]].
 
The [[Square root of a matrix|principal square root]] of an identity matrix is itself, and this is its only [[Positive-definite matrix|positive definite]] square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.<ref>Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of ''I''<sub>2</sub>". [[The Mathematical Gazette]] 87, November 2003, 499-500.</ref>
 
==See also==
*[[Logical matrix|Binary matrix]]
*[[Zero matrix]]
*[[Unitary matrix]]
 
==Notes==
<references />
 
==External links==
*{{planetmath reference|title=Identity matrix|id=1223}}
 
[[Category:Abstract algebra]]
[[Category:Linear algebra]]
[[Category:Matrices]]
[[Category:One]]
[[Category:Sparse matrices]]

Latest revision as of 02:39, 8 January 2015

Whenever you begin taking a look at new tires in your particular vehicle, you'll shortly notice ones which might be categorized as touring tires and people which are all season tires. So, what is the distinction between the 2? Is one better than the opposite? Here's a take a look at the solutions to those and a few other questions you may need.

It all sounds great and you are considering that this sounds to good to be true. This must be a rip-off or some marketer embellishing his success simply to make a sale. Well for the web marketing gurus it's not a rip-off or an embellishment. They know the key weapon to make all of this occur. Now they normally do not wish to tell anyone this secret, however I'm going to reveal it to you proper now. Financial institution Owned Used Work Truck, Vans Specials for Sale, Wholesale Pricing Financial institution owned used work vans specials are provided on the market with wholesale pricing via dealerships, auction houses, banks and other collaborating brokers. This Make sure there is no moisture trapped inside the tires that could rust rims and also cause pressure variations. Goodyear Tire Specs

The tread doesn't improve grip on the dry. A slick tire is more grippy because it places more rubber towards the pavement. Nevertheless, when a treaded tire will get worn, it signifies that the comfortable layer of rubber is shaved and the stiffer layers of rubber (used for structural rigidity) are uncovered. The tread also helps to disperse warmth, so bald tires are unhealthy. Nevertheless, it is vital to understand that fashionable tires do not wear out so quickely, they usually will have to be replaced long before the tread will get worn, because of other forms of damage.

Different shops usually have particular offers so its value checking regularly around the time you might want to exchange your tires. Other consideration needs to be given to discount golf equipment like Sams Membership and Costco but naked in thoughts you require a membership and these don't carry the large choice and experience of the aforementioned stockists. Ideas for discount Tires Or, perhaps you have got a extra fashionable tractor whose tires are all the identical measurement. In case you loved this article and you want to receive more details with regards to purchase of these tires kindly visit our own web page. No matter sort of tractor tires you're in search of, the main thing is to find a tire suited to your tractor at a value you can afford. Difference Between Winter Tires & All Season Tires You may wish to take into consideration this selection too. The place can you discover old tires that are okay to make use of? History of Tires Tips on how to Measurement Rims

One place you possibly can check the worth of tires is your native automotive restore shop. Most places that restore automobiles will also promote tires and it's possible you may discover the set that you just need on sale Typically a restore shop will run a promotion on a certain model of tires or a certain size. If there are several different restore shops in your area then chances are high you may have to confirm with all of them to see which one has the perfect deal.

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That rounds issues up on low cost tyres, you'll need to know the above prior to creating a sound investment on low-cost tyres whether or not its an on-line or off-line buy. One last point to recollect, should you do resolve to purchase some low cost car tyres then bear in mind that is quite a task to purchase a really dangerous set of tyres nowadays. Most tyres on the market are all safety checked before they can be legally bought. Simply be true to your requirements and get one of the best ones your pockets can safely afford. In regards to the Author

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