# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### Why can the determinant of a transformation matrix, and the original are be used to find out the new area?

Im studying Year 11 Mathematical Methods. Within the book there are several questions which give vertices (or sometimes just the area) of a particular shap, plus a transformation matrix to be applied ...
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### sign the elements of a ugly matrix

I have a matrix defined as $A=(I-aT)^{-1}F$, where I is identity matrix, a is a positive constant smaller than 1, T is a stochastic (transition matrix) and F is a matrix with positive diagonal ...
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### Derivation of why a diagonalizable matrix can be written as a sum of outer products $\Sigma=\sum_{i=1}^n \lambda_i v_i v_i^T$

Lets say we have a symmetric positive semi-definite $n\times n$ matrix $\Sigma$, which therefore has a diagonalization $\Sigma=V\Lambda V^T$, where $V$ is an orthogonal matrix (containing the ...
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### If $BNA=N$ it can happen that $B$ and $A$ identities?

Consider two matrices $A\in GL_m(K)$, $B\in GL_n(K)$ such that for any $N\in M_{n,m}(K)$ is true that $$BNA=N$$ How can I prove that $B$ and $A$ are identities?
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### Is square-root of a real symmetric positive semi-definite matrix real as well?

Given a real symmetric positive semi-definite matrix $A$, will there be a root $R$ which is real symmetric positive semi-definite as well? Can you comment on it's uniqueness? It will be nice if you ...
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### Solve linear system with $A_{i,j} = \langle e_i, e_j\rangle^2$, edges of a tetrahedron

I have six vectors in $e_i\in\mathbb{R}^3$ that are the edges of a tetrahedron. Let us consider now the linear equation system $Ax=b$ with  A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, ...
### If two real matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$? [duplicate]
As in the title: If two real (square) matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$?