What is Symmetric matrix: Definition and 63 Discussions
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if
a
i
j
{\displaystyle a_{ij}}
denotes the entry in the
i
{\displaystyle i}
th row and
j
{\displaystyle j}
th column then
for all indices
i
{\displaystyle i}
and
j
.
{\displaystyle j.}
Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
My attempt:
$$
\begin{vmatrix}
1-\lambda & b\\
b & a-\lambda
\end{vmatrix}
=0$$
$$(1-\lambda)(a-\lambda)-b^2=0$$
$$a-\lambda-a\lambda+\lambda^2-b^2=0$$
$$\lambda^2+(-1-a)\lambda +a-b^2=0$$
The value of ##\lambda## will be positive if D < 0, so
$$(-1-a)^2-4(a-b^2)<0$$
$$1+2a+a^2-4a+4b^2<0$$...
Hello,
I am often designing math exams for students of engineering.
What I ask is the following:
Can I choose any real 3x3 symmetric matrix with positive eigenvalues as a realistic matrix of inertia?
Possibly, there are secret connections between the off-diagonal elements (if not zero)...
Hello
Say I have a column of components
v = (x, y, z).
I can create a skew symmetric matrix:
M = [0, -z, y; z, 0; -x; -y, x, 0]
I can also go the other way and convert the skew symmetric matrix into a column of components.
Silly question now...
I have, in the past, referred to this as...
Homework Statement
Consider matrices A = [1 2;2 4] and P = [1 3;3 6]. Using B = P^-1*A*P, verify that similar matrices have the same eigenvalues. Find the eigenvectors y for B and show that x = P*y are eigenvectors of A.
Homework Equations
B = P^-1*A*P,
x = P*y
The Attempt at a Solution
I...
Homework Statement
Need to prove that the derivative of a rotation matrix is a skew symmetric matrix muktiplied by that rotation matrix. Specifically applying it on the Rodrigues’ formula.Homework EquationsThe Attempt at a Solution
I have shown that the cubed of the skew symmetric matrix is...
I split off this question from the thread here:
https://www.physicsforums.com/threads/error-in-landau-lifshitz-mechanics.901356/
In that thread, I was told that a symmetric matrix ##\mathbf{A}## with real positive definite eigenvalues ##\{\lambda_i\} \in \mathbb{R}^+## is always real. I feel...
I am currently brushing on my linear algebra skills when i read this
For any Matrix A
1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out )
2)(A + At)/2 is symmetric
Now my question is , why should it be...
Homework Statement
Hi there,
I'm happy with the proof that any odd ordered matrix's determinant is equal to zero. However, I am failing to see how it can be done specifically for a 3x3 matrix using only row and column interchanging.
Homework Equations
I have attached the determinant as an...
I'm looking for the general form of a symmetric 3×3 matrix (or tensor) ##\textbf{A}## with only two different eigenvalues, i.e. of a matrix with the diagonalized form ##\textbf{D}=\begin{pmatrix}a& 0 & 0\\0 & b & 0\\0 & 0 & b\end{pmatrix} = \text{diag}(a,b,b)##.
In general, such a matrix can be...
Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true.
True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.
Hi,
I'k looking at some MATLAB code specifically eig2image.m at:
http://www.mathworks.com/matlabcentral/fileexchange/24409-hessian-based-frangi-vesselness-filter/content/FrangiFilter2D
So, I understand how the computations are done with respect to the eigenvector / eigenvalues and using...
Hi,
Well I hope it's not a thread that already is in the storage here. I want to understand the image of a matrix. not only calculating it but also why I'm doing that. Here are my questions:
1) They say d = Lx has a solution if d ∈ ImL. I know that the image of a matrix is calculated by Lx = x...
Homework Statement
I try to run this program, but there are still some errors, please help me to solve this problems
Homework EquationsThe Attempt at a Solution
Program Main
!====================================================================
! eigenvalues and eigenvectors of a real...
Homework Statement
From Mary Boas' "Mathematical Methods in the Physical Science" 3rd Edition Chapter 3 Sec 11 Problem 33 ( 3.11.33 ).
Find the eigenvalues and the eigenvectors of the real symmetric matrix.
$$M=\begin{pmatrix} A & H \\ H & B \end{pmatrix}$$
Show the eigenvalues are real and...
I am a bit dense when it comes to linear algebra for some reason. I am reviewing math to prepare for a physics grad program, and I am using Mary Boas "Mathematical Methods in the Physical Sciences". She presents the idea of a skew symmetric matrix in the problem set rather than in the text. I...
I asked this question here, however the title of the thread (and the thread itself) was sloppy and unclear.I could not find a way to delete or edit.
This is for a regression analysis course, and I've only taken one introductory course on linear algebra, so when I Google'd "finding a symmetric...
Hello MHB.
During my Mechanics of Solids course in my Mechanical Engineering curriculum I came across a certain fact about $3\times 3$ matrices.
It said that any symmetric $3\times 3$ matrix $A$ (with real entries) whose trace is zero is orthogonally similar to a matrix $B$ which has only...
Homework Statement
We have a finite group ##G## and a homomorphism ##\rho: G \rightarrow \mathbb{GL}_n(\mathbb{R})## where ##n## is a positve integer. I need to show that there's an ##n\times n## positive definite symmetric matrix that satisfies ##\rho(g)^tA\rho(g)=A## for all ##g \in G##...
Hi everyone, :)
Here's a question I am stuck on. Hope you can provide some hints. :)
Problem:
Let \(U\) be a 4-dimensional subspace in the space of \(3\times 3\) matrices. Show that \(U\) contains a symmetric matrix.
I am asking for some hints to solve this excercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $ R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has the identity matrix in two...
Homework Statement
Generate a random 10 x 10 symmetric matrix A (already done in MATLAB) . Express A in the formHomework Equations
## A = ## \displaystyle \sum_{j=1}^{10}λ_j(A)v_jv^T_j\
for some real vectors ##v_j, j = 1, 2, . . . , 10.##
The Attempt at a Solution
I'm pretty sure the...
Hey guys if i have a vector x=[x1,x2, ... xn]
what are the eigenvectors and eigenvalues of X^T*X ?
I know that i get a n by n symmetric matrix with it's diagonal entries in
the form of Ʃ xii^2 for i=1,2,3,. . . ,n
Thank you in advance once again!
Hello to all of you,
Is there a way to get the matrix A=[a b c d] from the eigenvectors (orthogonal) matrix
H= sin(x) cos(x)
cos(x) -sin(x)
or to pose it differently to find a matrix that has these 2 eigenvectors ?
Thank you in advance .
Michael
Homework Statement
The question is:
Let C be a symmetric matrix of rank one. Prove that C must have the form C=aww^T, where a is a scalar and w is a vector of norm one.
Homework Equations
n/a
The Attempt at a Solution
I think we can easily prove that if C has the form...
The question is:Let $C$ be a symmetric matrix of rank one. Prove that $C$ must have the form $C=aww^T$, where $a$ is a scalar and $w$ is a vector of norm one.(I think we can easily prove that if $C$ has the form $C=aww^T$, then $C$ is symmetric and of rank one. But what about the opposite...
I have been trying to prove the following result:
If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors.
Is there a simple proof of this result?
I have been trying to prove the following result:
If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors.
Is there a simple proof of this result?
Eigen values of a complex symmetric matrix which is NOT a hermitian are not always real. I want to formulate conditions for which eigen values of a complex symmetric matrix (which is not hermitian) are real.
A 3x3 symmetric matrix has a null space of dimension one containing the vector (1,1,1). Find the bases and dimensions of the column space, row space, and left null space.
I understand how to get the Dim of Col(A), Row(A), and Nul(A^T) but how do i get the bases with just knowing the dimension...
Hello everyone!
I'm struggling to find a general formula for obtaining an inverse of a symmetric matrix, for e.g.
1 i -1
i -i 2
-1 2 1
Any help is appreciated!
Is it true that an nxn symmetric matrix has n linearly independent eigenvectors even for non-distinct eigenvalues? How can we show it rigorously? Basically, I want to prove that if an nxn symmetric matrix has eigenvalue 0 with multiplicity k, then its rank is (n - k). If we can prove that there...
Homework Statement
Given the matrix A:
4 2 2
2 4 2
2 2 4
Find the matrix P such that P-1AP is diagonal
Homework Equations
The Attempt at a Solution
So I had this question today on a placement exam and it threw me for a loop. I found the eigenvalues to be 2,2, and 8. The...
How to prove that the determinant of a symmetric matrix with the main diagonal elements zero and all other elements positive is not zero and different ?
First of all, I apologize if this is in the wrong place. I didn't really know where it should be placed and if it is in the wrong place I am sorry.
This question was on my recent Linear Algebra I final exam and I had no idea how to do it when I was writing the exam and I'm still stumped by...
Homework Statement
I need to diagonalize the matrix A=
1 2 3
2 5 7
3 7 11
The Attempt at a Solution
Subtracting λI and taking the determinant, the characteristic polynomial is
λ3 - 17λ2 + 9λ - 1 (I have checked this over and over)
The problem now is it has some ugly roots, none that I would...
Hello all!
I just had a question about combining elements of matrices.
In the MATLAB documentation, there was a function called triu and tril that extracts the upper and lower components of a matrix, respectively. I was wondering if there was a way to copy the elements of the upper triangle...
Homework Statement Let L(x) a linear operator defined by setting the diagonal elements of x to zero. What will be the representation of this operator to the following basis set? x E X. X denote the set of all real symmetric 3x3 matrices. Homework Equations
L*y=x
L=x*inv(y)...
I'm pretty inexperienced in proof writing. So not sure if this was valid.
If a matrix is skew symmetric then A^T = - A, that is the transpose of A is equal to negative A.
This implies that if A = a(i,j), then a(j,i) = -a(i,j). If we're referring to diagonal entries, we can say a(j,j) =...
Homework Statement
Let W be a 3x3 matrix where A^t(transpose)=-A. Find a basis for W.
Homework Equations
Find a basis for W.
The Attempt at a Solution
I have no idea how to start it.
Homework Statement
If A is a symmetric matrix, what can you say about the definiteness of A^2? Explain.
Homework Equations
I believe I need to use the face that A^2=SD^2S^-1.
I know that if all the eigenvalues of a symmetric matrix are positive then the matrix is positive...
Homework Statement
Hello,
I need some help in the fist parts of two lineal algebra problems, specially with algebraic manipulation. I guess that if I rewrite the determinant nicely some terms get canceled and I can write the inverse nicely, but don't know how to do it...
Problem 1...
Hi,
Is there a simplification for the determinant of a symmetric matrix? For example, I need to find the roots of \det [A(x)]
where
A(x) = \[ \left( \begin{array}{ccc}
f(x) & a_{12}(x) & a_{13}(x) \\
a_{12}(x) & f(x) & a_{23}(x) \\
a_{13}(x) & a_{23}(x) & f(x) \end{array}...
Homework Statement
Given a real diagonal matrix D, and a real symmetric matrix A,
Homework Equations
Let C=D*A.
The Attempt at a Solution
How to prove all the eigenvalues of matrix C are real numbers?
Help! Symmetric matrix
I know that all the eigenvalues of a real symmetric matrix are real numbers.
Now can anyone help out how to prove that "all the eigenvalues of a row-normalized real symmetric matrix are real numbers"? Thank you~~~