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Prove the inequality for Condition number of matrix

Inequality involving norm of matrix integralHow to show for a PSD matrix $A$ that $left | left ( A+I right )^-1 right | leq frac11+sigma _minleft ( A right )$?Condition number of a rectangular matrixRelation between the weighted matrix norm and the weightsRelative error in solution when the matrix is perturbedCondition number for a complex square matrixCondition Number and Sensitivity of Matrix-Vector MultiplicationA lower bound for the condition number matrixSuppose ||.|| is an induced matrix norm, A is non-singular, and B is singular. Prove $frac1kappa(A)leqfracA$.Prove that $cond(A)ge fracA$ for any induced matrix norm

1

$begingroup$

Let $ A in mathbbR^n times n$ be a non-singular matrix.

Let $hat A=A+delta A, hat x=x+delta x, textand hat b=b+delta b$ with $Ax=b$ and $hat A hat x=hat b $.

Here $||.||$ denote the both vector norm and induced matrix norm.

Show that $ large frac leq kappa (A) left(fracdelta AA+frac+frac right)$, where $kappa(A)=||A||||A^-1||. $

Help me to show it

matrices condition-number

share|cite|improve this question

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819

$endgroup$

add a comment | 

1

$begingroup$

Let $ A in mathbbR^n times n$ be a non-singular matrix.

Let $hat A=A+delta A, hat x=x+delta x, textand hat b=b+delta b$ with $Ax=b$ and $hat A hat x=hat b $.

Here $||.||$ denote the both vector norm and induced matrix norm.

Show that $ large frac leq kappa (A) left(fracdelta AA+frac+frac right)$, where $kappa(A)=||A||||A^-1||. $

Help me to show it

matrices condition-number

share|cite|improve this question

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819

$endgroup$

add a comment | 

$begingroup$

Let $ A in mathbbR^n times n$ be a non-singular matrix.

Let $hat A=A+delta A, hat x=x+delta x, textand hat b=b+delta b$ with $Ax=b$ and $hat A hat x=hat b $.

Here $||.||$ denote the both vector norm and induced matrix norm.

Show that $ large frac leq kappa (A) left(fracdelta AA+frac+frac right)$, where $kappa(A)=||A||||A^-1||. $

Help me to show it

matrices condition-number

share|cite|improve this question

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819

$endgroup$

share|cite|improve this question

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819

share|cite|improve this question

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

edited Mar 15 at 21:19

Rodrigo de Azevedo

13.2k41960

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819

asked Mar 15 at 18:50

M. A. SARKARM. A. SARKAR

2,4291819