# Questions tagged [symmetric-matrices]

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

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### How to solve $A^{\frac 12} B A^{\frac 12} = C$ for $A$?

Suppose that matrices $A,B,C$ are symmetric and positive definite. Then, $A$ has a unique, positive square root, which we call $A^{\frac 12}$. If $$A^{\frac 12} B A^{\frac 12} = C$$ then can we write ...
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### How to prove the transpose matrix is in a vector space with restrictions on the dimension

For an assignment in class, I have the following question. Let n $\geq 1$ and let W be a subspace of $Mn\times n(K)$ such that $dim(W)>\frac{n^2-n}{2}$. Prove that W contains a non-zero matrix ...
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### Show that A is skew-symmetric if and only if $x^tAx = 0$

I've tried by starting with setting $x^tAx = 0 = x^t(-A^t)x$ and checking it termwise, but I don't think this will show me anything. Could you explain how to approach this problem please?
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### Inverse of a matrix which is difference of a singular matrix with a small diagonal matrix?

If $A$ is a real symmetric singular matrix (similar to a Laplacian matrix, which comes from M'GM, where M is incidence matrix and G is a diagonal matrix) with large values and $B$ is a diagonal matrix ...
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### Cholesky factor when adding a row and column in between

I have a problem where I have the Cholesky factorization ($A=LL'$) of a symmetric positive-definite matrix. Now, I need to add a new row and column somewhere in the "middle" of the matrix and compute ...
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I have stumbled upon the following claim, and I wonder if it has a simple proof: Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\... 1answer 43 views ### Symmetric integer matrix and odd entry in the diagonal. K is a 2N$\times$2N symmetric integer matrix with at least one odd element in the diagonal. Suppose$\mathcal{M}$be a set of integer vectors satisfying the following two properties: 1)$m^{T}K^{-1}m'...
I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 &...