What is Diagonal matrix: Definition and 55 Discussions
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is
[
3
0
0
2
]
{\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}
, while an example of a 3×3 diagonal matrix is
[
6
0
0
0
7
0
0
0
4
]
{\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&7&0\\0&0&4\end{smallmatrix}}\right]}
. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix.
A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.
I came across a statement in《A First Course in General Relativity》:“The only matrix diagonal in all frames is a multiple of the identity：all its diagonal terms are equal.”Why?I don’t remember this conclusion in linear algebra.The preceding part of this sentence is:Viscosity is a force parallel...
I’m really unable to solve those questions which ask to find a nonsingular ##C## such that
$$
C^{-1} A C$$
is a digonal matrix. Some people solve it by finding the eigenvalues and then using it to form a diagonal matrix and setting it equal to $$C^{-1} A C$$. Can you please tell me from scratch...
First of all, it is clear that we can find such a bases (the theorem is given in almost all of the books, but if you want to share some insight I shall be highly grateful.)
We can show that ##W## will be the set of all real polynomials with degree ##\leq 2##. So, let's have ##\{1,x,x^2\}## as...
The following matrix is given.
Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1.
The answer sheet reads that the diagonal matrix D is as follows:
I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...
I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf
How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
Homework Statement
A = 000
010
101
Find Eigenvalues, its corresponding eigenvectors, and find a matrix W such that W^t*AW = D, where D is a diagnol matrix.(note that W^t represents the transpose of W)
Homework Equations
Eigenvalues, Eigenvectors, diagnolization[/B]The Attempt at a...
What does it mean by "In the position representation -- in which r is diagonal" in the paragraph below? How can we show that?
Does it mean equation (3) in http://scienceworld.wolfram.com/physics/PositionOperator.html? (where I believe the matrix is in the ##|E_n>## basis)
So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered
e^{At} \vec{u}(0) = \vec{u}(t)
as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices.
I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix.
The answer given is here, relevant answer is (b):
Imgur link: http://i.imgur.com/DKwt8cN.png...
Homework Statement
Is it necessary to arrange the eigenvalues in increasing value order?
As shown in the image attached, if I arrange my eigenvalues -2, -1, 1 diagonally, my D would be
2^8 , 1, 1 diagonally.
However if i arrange it as, say, -1, 1, -2, my D would be different...
Is it possible to prove the fact that any function of diagonal matrix is just a function of its element?
I don't know how I could express the proof. I can prove that a multiplication of diagonal matrix will just be the multiplication of its element using summation notation, or diagonal matrix...
Hey! So here's the question:
Homework Statement
Let
\mathbf{B} \in \mathbb{R}^{n \times n} be some square matrix we can choose and
\mathbf{D} \in \mathbb{R}^{n \times n} be some given diagonal matrix with positive diagonal elements.
For what matrices \mathbf{B} is the product...
Homework Statement
If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors?
Homework Equations
The Attempt at a Solution
Homework Statement
I'm solving an exercise about small oscillations.
I name T the kinetic matrix and $H$ the hessian matrix of potential.
The matrix \omega^2 T- H is diagonal and so find the auto-frequencies is easy! But I have a problem with normal modes. The lagrangian coordinates are two...
Apart from simplifying matrix powers, why do we want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices.
Thanks for any answers.
Apart from simplifying matrix powers, why do we want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices.
Thanks for any answers.
Homework Statement
Solve this system of differential equations
\begin{equation}
x'_1=5x_1 + 2 x_2 - x_3 \\
x'_2=-2x_1 + x_2 - 2x_3 \\
x'_3=-6x_1 - 6 x_2
\end{equation}
Homework Equations
The Attempt at a Solution
This is my first time solving a problem like this and I just...
Homework Statement
I am working on a problem where I made a matrix representation of a linear transformation and I am asked what is the eigenspace for a particular eigenvalue.
Homework Equations
The Attempt at a Solution
The problem for me is, I came out with a diagonal...
I'm not sure how to start this problem.
All i know is a diagonal matrix consists of all 0 elements except along the main diagonal.
But how do I even find a basis for this?
Homework Statement
I have
J=\begin{bmatrix}
\frac{\pi}{2}&0&0\\
1&\frac{\pi}{2}&0\\
0&1&\frac{\pi}{2}\\
\end{bmatrix}
I need to find \sin(J) \text{ and } \cos(J) \text{ and show that } \sin^{2}(J)+\cos^{2}(J)=I
Homework Equations
The Attempt at a Solution
I have the...
Homework Statement
Define L:R3-->R3 by L(x,y,z)=(y-z,x+z,-x+y).
A. Show that L is self-adjoint using the standard orthonormal basis B of R3.
B. Diagonalize L and find and orthogonal basis B of R3 of eigenvectors of L and the diagonal matrix.
C. Considering only the eigenvalues of L...
If you have a matrix that is a combination of a diagonal, upper, and lower matrix - what is the best way to solve it? (I use MATLAB for matrix work)
Example:
5 1 0 0 x1 1
-1 5 1 0 x2 2
0 -1 5 1 x3 3
0 0 -1 5 x4 4
Is it possible to solve...
I have two matrices which commute, one of which is definitely diagonal:
\textbf{B}diag\{\underline{\lambda}\} = diag\{\underline{\lambda}\}\textbf{B}
and I want to know what I can say about \textbf{B} and/or \underline{\lambda}. Specifically, I feel that either one or both of the following...
Diagonalizing an N × N matrix H involves writing it as H = UDU† where D is a
diagonal matrix, with diagonal elements equal to the eigenvalues of the matrix H, and U
is a unitary matrix.
We may write:
D=
(λ1 0 0 ... 0)
(0 λ2 0 ... 0)
(0 0 λ3... 0)
(... ... ... ... λn)...
Homework Statement
Write the A matrix and the x vector into a basis in which A is diagonal.
A=\begin{pmatrix} 0&-i&0&0&0 \\ i&0&0&0&0 \\ 0&0&3&0&0 \\ 0&0&0&1&-i \\ 0&0&0&i&-1 \end{pmatrix}.
x=\begin{pmatrix} 1 \\ a \\ i \\ b \\ -1 \end{pmatrix}.
Homework Equations
A=P^(-1)A'P.
The...
Homework Statement
A square matrix A (of some size n x n) satisfies the condition A^2 - 8A + 15I = 0.
(a) Show that this matrix is similar to a diagonal matrix.
(b) Show that for every positive integer k >= 8 there exists a matrix A
satisfying the above condition with tr(A) = k.
Homework...
I think this is a pretty simple question. I need a transformation that will take a Column vector e.g.: <a,b,c> and turn it into a 3x3 matrix where a is in position 1,1 and b in position 2,2 and c in position 3,3. i.e.: a diagonal matrix.
Any help?
Hi,
While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix D^{H}A^{H}AD where the matrix A is known and the matrix D is a diagonal matrix whose entries contain the variables that need to be optimized (those...
Homework Statement
If we have a n x n matrix with 1 on the diagonal entries apart from the ith column which has a -1. As well as this ith row can have any real number in each entry. Other than this the matrix is 0 everywhere.
Show this matrix is diagonalisable.
Homework Equations...
Consider two matrices:
1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n;
2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n.
If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...
Suppose I have a matrix that I want to reduce to block diagonal form. Obviously, the block diagonal form is not unique as each of the diagonal blocks is defined only to within a unitary rotation. So I want to find the block diagonal matrix that is closest to the original matrix in terms of the...
Homework Statement
center of the general linear group is diagonial matrix proof
Homework Equations
The Attempt at a Solution
i write out a n by n matrix and multiply left by a and right by a^-1 and show that it is the same.
I think it can force the matrix to be diagonal but i...
Homework Statement
well, my assignment was to make a gauss elimination, so now i need to compute the determinant of an n by n diagonal matrix
variable rows = number of equations
variable i = random integer
matrix A[100][100]
dummy matrix A2[100][100]
Homework Equations
det[A]...
Homework Statement
A= [1 -1 0]
[-1 2 -1]
[0 -1 1]
find orthogonal matrix P and diagonal matrix D such that P' A P = D
Homework Equations
The Attempt at a Solution
i got eigenvalues are 0, 1, 3 which make D=[0 0 0; 0 1 0; 0 0 3]
how to find P. because in...
Homework Statement
If A=[{5,3},{-2,-2}], find the eigenvectors of A. Using these eigenvectors as matrix P, find P-1 and thus prove P-1AP is diagonal.
Homework Equations
None
The Attempt at a Solution
So i can get the eigenvectors to be <3,-1> and <1,-2> corresponding to eigenvalues 4...
Homework Statement
Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1
a) Show that D is idempotent
b) Show that if X is a nonsingular matrix and A=XD(X)-1 , then A is idempotentHomework Equations
The Attempt at a Solution
a) I tried it, and it works for a specific...
Homework Statement
Let T be the linear operator on R3 that has the given matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. a) Determine whether T can be represented by a diagonal matrix, and b) whenever possible, find a diagonal matrix and a basis of R3 such that T is represented...
Hi,
I have been struggling with this problem for a while, and I have not found the answer in textbooks or google. Any help would be very much appreciated.
Suppose I know the singular value decomposition of matrix B, which is a singular, circulant matrix. That is, I know u_i, v_i, and...
Homework Statement
Find an invertible matrix P and a diagonal matrix D such that D=P^(-1)AP.
A=
(−1 2 −2)
(−2 3 −2)
( 2 −2 3)
Homework Equations
The Attempt at a Solution
the eigenvalues are 1, 1, and 3
the eigenvector I've found so far is for the eigenvalue 3...
Hi I have this question for my Linear Algebra class and I can't seem to figure it out.
Let A and B be n x n matrices such that B = (P^-1)AP and let lambda ne an eigenvalue of A (and hence of B). Prove the following results:
(a) A vector b in R^n is in the eigenspace of A corresponding to...
Homework Statement
Is The space P2 is isomorphic to the space of all 3 × 3 diagonal matrices.
Homework Equations
The Attempt at a Solution
I know that P2 is isomorphic to vectors with 3 components so i think this statement is true, is it?
Homework Statement
Let f(x,y,z)=(exp(x),cos(y),sin(z)).Compute the Jacobian J(f) of f . In general ,when will the Jacobian J(g) of a function g(x,y,z) be a diagonal matrix ?
Homework Equations
The Attempt at a Solution
I am not quiet sure about this question
for J(f) i found...
Whats the physical reason for a diagonalized matrix with the eigenvalues of the system as the values?
Reading from wiki, it seems that its something to do with the Schrodinger Eqn, but I don't follow that. If someone could explain the point of it to me in plain English (or as close as it...
Homework Statement
Homework Equations
For a symmetric matrix B=B' where ' is the transpose.
The Attempt at a Solution
Since we know that for a symmetric matrix,
B = B'
I attempted to substitude that in and tried to solve for D.
DTD-1 = (D-1)'T'D
DT = D-1T'DD
D =...
I am trying to create the following Matrix without the use of a for loop.
1 2 1 0 0 0 0 0 0 0 0 0
0 0 1 2 1 0 0 0 0 0 0 0
0 0 0 0 1 2 1 0 0 0 0 0
0 0 0 0 0 0 1 2 1 0 0 0
Etc.
Is there a simple way of doing this
Edit: Using Matlab
If V is diagonal, it is easy to show:
(V + V^{-1})^{-1} = V(V^2 + I)^{-1}
by multiplying both sides by:
(V + V^{-1})
But, I'm wondering if there is a way to derive the RHS from the LHS. Since diagonal matrices behave like scalars, I used a scalar analogy:
(x + 1/x)^{-1} = ((x^2...