# 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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### Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
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### Operator norm and eigenvalue inequality

Can I say that $\|A\| < s$ where $A \in \mathbb{R}^{3 \times 3}$ is a symmetric, positive definite matrix and $s$ is the maximum eigenvalue of $A$. Here the norm used is operator norm.
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### Hamming's code is perfect

How does one prove that Hamming's code is perfect (i.e. it is the 1-error correcting code that has the smallest possible size). I haven't found a complete proof using Google.
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### Determinant of a block rectangular matrix

Assume that $a \in \mathbb{R}^{n \times 1}, b \in \mathbb{R}, P \in \mathbb{R}^{n \times n}, u \in \mathbb{R}^{n \times 1}, x \in \mathbb{R}^{n \times 1}$ and $\lambda \in \mathbb{R}$. Now I want to ...
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### Understanding Ax = 0 in Linear Algebra

If we have 3 systems of equations, that intersect at the point (1,2,3) does it have trivial or nontrivial solutions? Let's assume it is this system of equations which intersects at (1,2,3) and row ...