# Questions tagged [symmetric-matrices]

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

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### Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$\mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k},$$ ...
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### Is a scalar product positive definite on a unique maximal subspace?

Let $V$ be an $n$-dimensional real vector space and \begin{equation} \eta\colon V\times V\to\mathbb R \end{equation} a nondegenerate symmetric bilinear form. Sylvester’s Law of Inertia allows us to ...
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### Euclidean projection on convex set of positive semidefinite matrices

Define the Euclidean projection for a convex set $C$ as follows $$\pi_C(y) := \min_{x \in C} \| y - x \|_2^2$$ How would we find the projection map when $C$ is the cone of positive semidefinite ...
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### Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

In a linear algebra textbook, I was given the following problem: If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric. I know that there are relatively ...
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### Decomposition of order 3 tensor symmetric along two dimensions

I have a 3rd order tensor $\mathbf{A}$ consisting of symmetric covariance matrices (with dimensions of space by space) stacked in time. I would like to compute the leading spatial features that ...
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### If $A$ is a symmetric matrix, then $\det(A) \leq \prod\limits_{i = 1}^d a_{ii}.$

Prove or provide a counterexample. If $A = (a_{ij})$ is a symmetric matrix, then $\det(A) \leq \prod\limits_{i = 1}^d a_{ii}.$ The result is obviously true for diagonal matrices and here is a proof ...
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### Universal Properties of Orthogonal Matrices

I wanted to ask this question since I have seen conflicting viewpoints on it. Are orthogonal matrices necessarily symmetric? I do not believe so but some website said they were so I need to confirm. (...
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### The square of a real skew-symmetrix matrix is symmetric matrix

Let $S \in \mathcal{M}_n(\mathbb{C})$ be a real symmetric matrix. Show that $S$ is the square of a real skew-symmetrix matrix then all its eigenvalues are negative and its non-positive eigenvalues ...