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.

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Condition number of a product of two matrices

Given two square matrices $A$ and $B$, is the following inequality $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number, ...
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1answer
263 views

Is Schur complement better conditioned than the original matrix?

Consider the following linear system (in block form) with s.p.d. matrix: $$ \begin{pmatrix} A & B\\ B^\top & C \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} f\\g \end{...
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3answers
532 views

How to measure how far a matrix is from being singular?

What would be the best mathematical tool/concept to measure how far a matrix is from being singular? Could it be the condition number?
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1answer
859 views

Matrix condition number and loss of accuracy

There are quite a few sources online that say something along the lines of : "As a rule of thumb, if the condition number $\kappa(A)=10^k$ then you may lose up to $k$ digits of accuracy on top of ...
2
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1answer
159 views

Condition number of matrix is $1$ iff $A^TA=\alpha I$

I am trying to prove the following: The condition number $\kappa_p(A)=1$ for $A$ an matrix iff $A^TA=\alpha I$ for some scalar number $\alpha\neq 0$. I read somewhere on the internet that I ...
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3answers
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Sensitivity of the least squares method and matrix condition number

It is my understanding that if we have a simple linear system such as $$Ax = y$$ The condition number of $A$ provides an indication of how sensitive the solution $x$ will be in relation to changes ...
1
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3answers
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multiplying a linear system by an invertible diagonal matrix

let $Ax = b$ be a linear n by n system if we multiply this system by a non-singular diagonal $D$ I can say that the new system still has the same solution as the previous one , right ? I ran some ...
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1answer
405 views

Relative error in solution when the matrix is perturbed

Suppose that $A \in \mathbb{R}^{n \times n}$ is an invertible matrix and that $Ax=b$ for some $x,b \in \mathbb{R}^{n}$. If $\kappa(A)$ denotes the condition number of $A$, it is well known that if $b$ ...
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0answers
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Consistent proxy for MSE in Least Squares

Suppose I have two $N \times M$, $N \gg M$ matrices $\mathbf{A}$ and $\mathbf{B}$. Both have a structure and are generated using an analytical description from a set of iid random variables $X_i$. ...
0
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1answer
126 views

Condition number is less than n

Show that for an $n$ x $n$ orthogonal matrix $A$ that $\operatorname{Cond}(A) \leq n$. I need to use: $$\|x\|_1 \leq \sqrt n$$ I know that $\operatorname{Cond}(A)=1$ for $A$ orthogonal matrix. ...