Questions tagged [condition-number]

The condition number of a matrix is the ratio of the largest to the smallest singular value in the singular value decomposition of a matrix.

128 questions
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Prove that $\kappa_F(A) \ge \sqrt{n}$

I reached at a point after solving it $$\kappa_F(A) = \sqrt{\mbox{tr}(A^H A) \cdot \mbox{tr}(A^{-1}(A^{-1})^H}$$ Now I am stuck. How to proceed? Or, alternatively, a new approach is also appreciated....
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1/Infinite-norm condition number and the QR decomposition

I have seen that often the condition number of a matrix, $A$, may be estimated by taking the QR decomposition of $A$, $A=QR$, and using numerical methods to estimate the condition number of the upper ...
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Use SVD to reduce condition number of matrix

I need to compute the inverse of an ill-conditioned matrix. Since condition number is ratio of high/low singular values. I am approximating the matrix by removing small singular values. But the ...
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Condition number of a matrix-vector product

If $A$ is an $m \times n$ matrix and $x$ is an $n \times 1$ vector then the linear transformation $y=Ax$ maps $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, so the linear transformation should have a ...
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Calculating condition number using by infinite norm of a vector [duplicate]

I need to calculate the condition number of A*x using infinity norm but ı know that ı can't use infinite norm for calclulating condition number.Am ı right?Could you please help me for this question ...
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Is the condition number of a 2x2 block symmetric matrix greater than the condition number of its upper left hand block?

Is there any known relation between cond(M) and cond(Q) when $$M=\begin{bmatrix}Q&A^T\\A&0\end{bmatrix}$$ and Q is symmetric positive definite and A is rectangular full row rank? From the ...
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Is division ill-conditioned when divisor is close to zero?

My intuition is that division of real numbers is ill-conditioned when divisor is close to zero. Is this intuition correct?
what is the significance of $\kappa$ in the proof of inconditional stability of the Crank-Nicolson scheme
When using a centered difference approximation $$\frac{\partial}{\partial t}u(t,x) = \frac{u(t + \Delta t/2,x) - u(t - \Delta t/2, x)}{\Delta t} + O((\Delta t)^2)$$ It is an approximation of the ...