Questions tagged [symmetric-matrices]

A symmetric matrix is a square matrix that is equal to its transpose.

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $A^T=A$.

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ commute, i.e., if $AB = BA$. So for integer $n$, $A^n$ is symmetric if $A$ is symmetric. If $A^{−1}$ exists, it is symmetric if and only if $A$ is symmetric.

The complex generalization is a hermitian matrix, a square matrix equal to its conjugate transpose. This is often denoted $A=A^{H}$ or $A=\overline{A^T}$; see for more information.

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Positive Definiteness of a symmetrized matrix involving the Moore Penrose Pseudoinverse

Let $M\in\mathbb{R}^{m\times n }$ and $V\in\mathbb{R}^{n\times l }$ both have full column rank and let $X^{+}$ denote the Moore Penrose Pseudoinverse of the matrix $X$. Question: Is the symmetrized matrix…
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which of the following are the rotation matrices?

which of the following are the rotation matrices? I know that if det|matrix|=1 or M.Mt=I=Mt.M is the property that makes some matrix a rotation matrix. but in this case options (1,3,4) seems to be alright as there determinant is 1 and above…
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Why is the computation of the general $p$-norm hard?

Consider a $2 \times 2$ matrix. Can we find a general polynomial for computing the operator norm induced by a $p$-norm?
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Symmetric, real, invertible matrix: How to prove component multiplication equals Kronecker?

Im currently looking for a prove. Given a real and invertible matrix $M$ with $M=M^T$, I would like to prove $M_{ij} M_{jk}^{-1}=\delta_{ik}$, where $\delta_{ik}$ is the Kronecker-delta (defined by $\delta_{i=k}=1$ and $\delta_{i\neq k}=0$).
Kartast
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$\forall C\in \mathbb{R}^{n\times n}, \ \ \ PAP-CMC^T\geq 0\ \ \ \Longrightarrow\ \ \ A-CMC^T\geq 0$

let M be a positive definite $n\times n$ matrix and A a positive semidefinite $n\times n$ matrix and P is an orthogonal projector of some subspace of $\mathbb{R}^n$ into $\mathbb{R}^n$ so is this implication correct $$\forall C\in…
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Can you help me with the 4th option?

I have solved for options 1, 2 and 3 but I'm stuck at the 4th one. Can anyone help?
Asad Ahmad
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Prove or disprove, all eigenvalues of a real symmetric matrix are non-negative.

I tried to find an answer for this question, but what I found was a classification of general matrices (i.e. definite, semi-definite and indefinite). I want to know more specifically about symmetric matrices.
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Do hermitian matrices commute when they occupy they same elements but have different values?

Given hermitian matrices A and B, they have different values but share the same non zero elements, e.g. $A=\begin{pmatrix}1&0&3\\0&2&4\\3&4&7\end{pmatrix}$ and $B=\begin{pmatrix}5&0&9\\0&7&1\\9&1&3\end{pmatrix}$ I am not familiar with the correct…
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Which of the following statements about determinants are correct?

Which of the following statements about determinants are correct? $\det(A^2)>0$, for all invertible matrices $A$ $\det(A+A^{-1})=\det(A)+\dfrac{1}{\det(A)}$, for all invertible matrices $A$ $\det(vv^T)>0$, for all column vectors $v ≠…
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Finding and creating Symmetric Matrix

Given the matrix, find symmetric closure of it. I am having a hard time understanding how to solve this. \begin{bmatrix}1&0&1&0&0\\1&1&0&1&1\\0&0&0&0&0\\0&1&1&0&0\\1&0&0&1&1\end{bmatrix} my try [ 0 1 1 1 1   1 0 1 0 1   1 1 1 1 1   1 0 1 1 1    1 1…
Jona
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A simple formula about matrix needed proof

I have a very simple question, but I do not know if I am right. Suppose we have a $n\times n$ real symmetric matrix $A$, i.e., $A=A^T$ and also we know that $A$ is invertible. Provided with a $n\times 1$ vector $b$, will we have $Abb^TA^{-1}=bb^T$ ?…
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Diagonalisation of symmetric Matrix

Suppose we have to find a diagonal matrix similar to a symmetric matrix... Is it possible to just have the diagonal matrix which can be generated by congruent operations and say the matrix is similar to the diagonal matrix generated by the congruent…
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Let $A$ and $B$ be matrix symmetric matrix. Show that $AB-BA$ is a skew symmetric matrix.

I solved it by myself, but I don't know I did it right. I hope you guys to check this out. Suppose $(AB-BA)^{T}=BA-AB$ $(AB-BA)^{T}=B^{T}A^{T}-A^{T}B^{T}$ then, $B^{T}=B$ $A^{T}=A$ therefore, $B^{T}A^{T}-A^{T}B^{T}=BA-AB$ It's my first time to ask…
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Sufficient condition for matrix $A$ to be similar to symmetric matrix $B$

Perhaps I'm missing something obvious, but what are sufficient conditions for a matrix $A$ to be similar to a symmetric matrix $B$? For instance, on the Wikipedia page for symmetric matrices, I see something about when a matrix is "symmetrizable",…
Uthsav Chitra
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I don't understand what this question mean, but I need to prove it for homework

Let A be a real symmetric matriz that σ(A) = {0}. Prove that A=O This exercise is for orthogonal diagonalization. How do I apply the spectrum theorem to get this proof?
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