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Expert-verifiedFound in: Page 267

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**Show that \(I - A\) is invertible when all the eigenvalues of \(A\) are less than 1 in magnitude. (Hint: What would be true if \(I - A\) were not invertible?)**

It is proved that \(I - A\) is invertible when all the eigenvalues of \(A\) are less than 1 in magnitude.

The eigenvalue \(\lambda \) is the real or complex number of a matrix \(A\) which is a square matrix that satisfies the following equation

\(\det \left( {A - \lambda I} \right) = 0\).

This equation is called the characteristic equation.

** **

If \(I - A\) is not invertible, then the equation\(\left( {I - A} \right){\rm{x}} = 0\) would have a non-trivial solution of \({\rm{x}}\).

Then \({\rm{x}} - A{\rm{x}} = 0\) and \(A{\rm{x}} = 1 \cdot {\rm{x}}\), which shows that \(A\) would have \(1\) as an

eigenvalue.

This cannot happen if all the eigenvalues are less than \(1\)in magnitude.

So \(I - A\) must be invertible.

It is proved that \(I - A\) is invertible when all the eigenvalues of \(A\) are less than 1 in magnitude.

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