What is Canonical form: Definition and 40 Discussions
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and which allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example:
Jordan normal form is a canonical form for matrix similarity.
The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix.In computer science, and more specifically in computer algebra, when representing mathematical objects in a computer, there are usually many different ways to represent the same object. In this context, a canonical form is a representation such that every object has a unique representation (with canonicalization being the process through which a representation is put into its canonical form). Thus, the equality of two objects can easily be tested by testing the equality of their canonical forms.
Despite this advantage, canonical forms frequently depend on arbitrary choices (like ordering the variables), which introduce difficulties for testing the equality of two objects resulting on independent computations. Therefore, in computer algebra, normal form is a weaker notion: A normal form is a representation such that zero is uniquely represented. This allows testing for equality by putting the difference of two objects in normal form.
Canonical form can also mean a differential form that is defined in a natural (canonical) way.
Hello:
I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables:
$$ax^2+by^2+cxy+dx+ey+f=0$$
Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part):
$$ w^TDw+[d \ \ e]w+f=0$$
$$w^TDw+Lw+f=0$$
where
$$...
Hi,
I found the above observable canonical form using this source:
https://www.mathworks.com/help/control/ug/canonical-state-space-realizations.html#mw_a76b9bac-e8fd-4d0e-8c86-e31e657471cc
I'm almost certain that I did do it correctly. But the code below gives me different values for B, C...
Hi,
I was recently being taught a control theory course and was going through a 'derivation' of the controllable canonical form. I have a question about a certain step in the process.
Question: Why does the coefficient ## b_0 ## in front of the ## u(t) ## mean that the output ## y(t) = b_0 y_1...
I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes:
Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its canonical form. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1...
Hello everyone,
I actually had a problem with understanding the part where they have defined L'AL = Λ. There, they have taken
γΛγ1 = Σy2λ = 1. Why have they taken that? Is it arbitary or does it come as a result of a derivation?
Thank you
Homework Statement
About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know.
$$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$
Find the
a) Jordan canonical form of ##A##
b) characteristic polynomial
c) minimal polynomial
d) ##dim\,kerA##
When:
case 1: we know that ##A## is...
Homework Statement
Find the Jordan canonical form of the matrix ##
\left( \begin{array}{ccc}
1 & 1 \\
-1 & 3 \\ \end{array} \right)##.
Homework EquationsThe Attempt at a Solution
So my professor gave us the following procedure:
1. Find the eigenvalues for each matrix A. Your characteristic...
Problem:
Express the quadratic form:
q=x1x2+x1x3+x2x3
in canonical form using Lagrange's Method/Algorithm
Attempt:
Not really applicable in this case due to the nature of my question
The answer is as follows:
Using the change of variables:
x1=y1+y2
x2=y1-y2
x3=y3
Indeed you get...
Homework Statement
For each matrix A, I need to find a basis for each generalized eigenspace of ## L_A ## consisting of a union of disjoint cycles of generalized eigenvectors. Then I need to find the Jordan canonical form of A.
The matrices are:
## a)
\begin{pmatrix}
1 & 1\\
-1 & 3...
Hello! (Wave)
A linear programming problem is in canonical form if it's of the following form:
$$\pm \max (c_1 x_1+ \dots + c_n x_n) , c_1, \dots, c_n \in \mathbb{R} \\ Ax=b, A \in F^{m \times n}, x=\begin{bmatrix}
x_1\\
\dots\\
\dots \\
x_n
\end{bmatrix}, b=\begin{bmatrix}
b_1\\
\dots\\...
I just finished a course on linear algebra which ended with Jordan Canonical Forms. There were many statements like "Jordan canonical forms are extremely useful," etc. However, we only learned a process to put things into Jordan canonical form, and that was it.
What makes Jordan canonical...
Normal form of the hyperbolic equation
Hey! :o
I am looking at the following in my notes:
$$a(x,y) u_{xx}+2 b(x,y) u_{xy}+c(x,y) u_{yy}=d(x,y,u,u_x,u_y)$$
$$A u_{\xi \xi}+ 2B u_{\xi \eta}+C u_{\eta \eta}=D$$
$$A=a \xi_x^2+2b \xi_y \xi_x+c \xi_y^2 \ \ \ (*)$$
$$B=a \xi_x \eta_x +b \eta_x...
the actual problem is to show that
the given matrix is similar to companion matrix
here is the companion matrix
Companion matrix - Wikipedia, the free encyclopedia
----------------
i know that if same frobenius canonical form then similar but i don't even know how to find the frobenius...
Hi everyone, :)
Take a look at the following question.
Problem:
Determine which of the following quadratic functions \(q_1,\,q_2:\,V\rightarrow\Re\) is positive definite and find a basis of \(V\) where one of \(q_1,\,q_2\) has normal form and the other canonical...
Homework Statement
I want to find the transition matrix for the rational canonical form of the matrix A below.
Homework Equations
The Attempt at a Solution
Let ##A## be the 3x3 matrix
##\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}##
The...
I have to prove the following result:
Let A,B be two n×n matrices over the field F and A,B have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than 3, then A and B are similar.
I have to use the following result:
If A,B are...
I'm trying to put a matrix into RCF, and I keep running into problems. I've checked my work a few times, so I think I must be making a conceptual error. Here's what I've got: $$A=\left( \begin{matrix}2 & -2 & 14 \\ 0 & 3 & -7 \\ 0 & 0 & 2\end{matrix}\right)\quad \text{ so }\quad xI-A=\left(...
I am trying to make this into jordan canonical form.
How can I box in the bottom two lambdas?
$$
\left[\begin{array}{ccc}
\begin{array}{cccc|}
\lambda & 1 & 0 & \\
& \lambda & 1 & 0\\
& & \lambda & 1\\
& & & \lambda\\\hline
\end{array}
& & \\
& \begin{array}{c|}
\lambda\\\hline
\end{array}...
As I understand, a Markov chain transition matrix rewritten in its canonical form is a large matrix that can be separated into quadrants: a zero matrix, an identity matrix, a transient to absorbing matrix, and a transient to transient matrix.
The zero matrix and identity matrix parts are easy...
Okay I want to be clear, this is not an homework question.
But I have a hard time understanding the concept of what I'm going to show you. The teacher goes way to fast for me, honestly, and when we ask a question it's just like he doesn't want to take the time to make us understand, kinda...
Homework Statement
Given the invariant factors of the matrix λI - A are
d1 to d5 = 1, d6 = (λ-3)(λ-2)2 , d7 = (λ-3)2(λ-2)2
Display the rational canonical form of A
Display the Jordan canonical form of A
The Attempt at a Solution
I'm half guessing here, but we have for d6 : λ3 - 7λ2 +...
could someone please explain briefly what the problem is with my method of finding such canonical forms.
The method we've been taught is to find the canonical form by performing double row/column operations on the matrix representation of quadratic form until we get to a diagonal matrix, and...
Hi, I'm just new here, I don't know if I'm on the right thread.:D
Homework Statement
Let F be a subfield of K. A, B be elements of Mn(F). Show that if A and B are similar over K, then A,B are similar over F. (Hint: what can be said about the rank of f(C(f(x)^m))^n? about the rank of...
Homework Statement
Let T:V->W be a linear transformation. Prove that if V=W (So that T is linear operator on V) and λ is an eigenvalue on T, then for any positive integer K
N((T-λI)^k) = N((λI-T)^k)
Homework Equations
T(-v) = -T(v)
N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all...
Homework Statement
Hi all.
I have no clue on how to do this problem because I missed the class where he covered this so could someone please walk me through it.
A = [2 2 -5; 2 7 2; -5 -15 -4] where ; means new column
p(x) = (x-3)(x-1)^2
1) what are the choices of m(x)
2) find...
Hello! I'm trying to do some linear algebra. I have an insane Russian teach whose English is, uh, lacking.. so I'd appreciate any help with these I can get here!
Homework Statement
Find the canonical forms for the following linear operators and the matrices for the corresponsing change of...
Hi, can someone show me how one would go about finding the matrix Q in Q^(-1) A Q = RationalCanonicalForm(A). Please demonstrate using the example
{4, 1, 2, 0}
{-4, 0, 1, 5} = A
{0, 0, 1, -1}
{0, 0, 1, 3}
where the characteristic polynomial is (x-2)^4 and the minimal polynomial is...
two part question to which i have answered the first part and am stuck on the 2nd part
find r and invertible real matrices Q and P such that Q-1AP=(Ir,0),(0,0)
where each 0 denotes a matrix of zeroes(not necessarily the same size in each case)
second part being paying special attention to...
Does anybody know of any good websites that contain a clear proof of the existence of the Jordan Canonical Form of matrices? My professor really confused me today
I've followed and understood this small example of calculating jordan forms all the way to the last line where they say "Therefore, the jordan form is...". When they say "therefore", it's NEVER obvious :smile:
Anyway, I get why the diagonal entries are -1. And that a minimal polynomial (t+1)^2...
1. Show that two matrices A,B ∈ Mn(C) are similar if and only if they share a Jordan canonical form.
2. Prove or disprove: A square matrix A ∈ Mn (F) is similar to its transpose AT. If the statement is false, ﬁnd a condition which makes it true.
(I'm pretty sure that this is true and can be...
I'm trying to teach myself math for physics (a middle aged physicist wannabee). Wikipedia's proof for the exisitence of a JC form for matrix A in Cn,n states:
"The range of A − λ I, denoted by , is an invariant subspace of A"
I'm having trouble seeing why any element of Ran(A − λ I) is in...
Homework Statement
Matrix:
\left| \begin{array}{ccc}
\-1 & -2 & 5 \\
6 & 3 & -4 \\
-3 & 3 & -11 \end{array} \right|\]
Homework Equations
The Attempt at a Solution
How will this matrix transferred into canonical form? What is actually canonical form?
How do I transform a second-order PDE with constant coefficients into the canonical form?
I tried to solve this problem:
u_xx + 13u_yy + 14u_zz - 6u_xy + 6u_yz + 2u_xz -u_x +2u_y = 0
I wrote the bilinear form of the second order derivatives and diagonalized it. I found out that it is a...
Or something like that... I need definition,, explanation and examples. I have an exam in Rings and Fields on Sunday, and he used that term during the course- I have no idea what it is.
I'd appreciate any help.
Thanks in advance!