# 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.

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1. ### Condition such that the symmetric matrix has only positive eigenvalues

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 $$(-1-a)^2-4(a-b^2)<0$$ $$1+2a+a^2-4a+4b^2<0$$...
2. ### Can every symmetric matrix be a matrix of inertia?

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)...
3. ### I Un-skewing a skew symmetric matrix (for want of a better phrase)

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...
4. ### Are Similar Matrices' Eigenvalues the Same? Solving for Symmetric Matrices

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...
5. M

### What is the derivative of a skew symmetric matrix?

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...
6. ### I Is a symmetric matrix with positive eigenvalues always real?

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...
7. ### I Linear algebra ( symmetric matrix)

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...
8. ### I Relationship Between Hermitian and Symmetric Matrices

Are All symmetric matrices with real number entires Hermitian? What about the other way around-are all Hermitian matrices symmetric?
9. ### Prove 3x3 Skew symmetric matrix determinant is equal to zero

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...
10. ### I General form of symmetric 3x3 matrix with only 2 eigenvalues

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...

17. ### Understanding Skew Symmetric Matrices for Physics - A Helpful Guide

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...
18. ### Find the unique symmetric matrix A such that Y'AY=Y'GY

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...
19. ### What is the minimum rank of a skew symmetric matrix?

What is minimum possible rank of skew symmetric matrix ?
20. ### MHB Zero-Trace Symmetric Matrix is Orthogonally Similar to A Zero-Diagonal Matrix.

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...
21. ### Positive-definite symmetric matrix satisfying a certain property

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##...
22. ### MHB The Existence of Symmetric Matrices in Subspaces

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.
23. ### How can I diagonalize this symmetric matrix?

\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{array}
24. ### Solve Invertible Skew Symmetric Matrix: Hints & Tips

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...
25. ### Expressing a symmetric matrix in terms on eigenvalues/vectors

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...
26. ### Eigen values and Eigenvectors for a special case of a symmetric matrix

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!
27. ### Find the symmetric matrix from eigen vectors

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
28. ### Form of symmetric matrix of rank one

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...
29. ### MHB Form of symmetric matrix of rank one

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...
30. ### MHB Repeated eigenvalues of a symmetric matrix

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?
31. ### Repeated eigenvalues of a symmetric matrix

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?
32. ### Eigenvalues of a complex symmetric matrix

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.
33. ### Linear Algebra (Symmetric Matrix)

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...
34. ### MHB Finding the Inverse of Symmetric Matrices with Non-Real Coefficients

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!
35. ### Eigenvectors of a symmetric matrix.

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...
36. ### Diagonalizing a symmetric matrix with non-distinct eigenvalues

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...
37. ### Determinant of symmetric matrix with non negative integer element

Let \begin{equation*} A=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & n-2 & \cdots & 0 & 1 \\ n& n-2 & \cdots & 1 & 0% \end{bmatrix}% \end{equation*}. How can you prove that det(A)=[(-1)^n][n][2^(n-1)]? Thanks.
38. ### Determinant of Symmetric Matrix

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 ?
39. ### Linear Algebra Symmetric Matrix Set Question

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...
40. ### Is there a way to diagonalize a symmetric matrix without using a calculator?

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...
41. ### MATLAB [Matlab]Copy Lower Triangle of symmetric matrix to Upper Triangle(or visa versa)

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...
42. ### Proving the Even Rank of Skew Symmetric Matrices: Induction and Other Methods

how can we prove that the rank of skew symmetric matrix is even i could prove it by induction is there another way
43. ### Calculating Representation of Linear Operator for Symmetric Matrix

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)...
44. ### Show that diagonal entries of a skew symmetric matrix are zero.

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) =...
45. ### Basis of skew symmetric matrix

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.
46. ### Symmetric Matrix and Definiteness

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...
47. ### Lineal Algebra: Inverse Matrix of Symmetric Matrix

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...
48. ### Determinant of a symmetric matrix

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}...
49. ### Proving Real Eigenvalues for Symmetric Matrix Multiplication?

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?
50. ### Help Prove Real Eigenvalues of Symmetric Matrix

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~~~