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Let \(A\) be the rank one \(n \times n\) matrix \(A=\left(v_{i} v_{j}\right)\), where \(\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)\) is a non-zero real vector.


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Let \(A\) be the rank one \(n \times n\) matrix \(A=\left(v_{i} v_{j}\right)\), where \(\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)\) is a non-zero real vector.

a) Find its eigenvalues and eigenvectors.
b) Find the eigenvalues and eigenvectors for \(A+c I\), where \(c \in \mathbb{R}\).
c) Find a formula for \((I+A)^{-1}\). [ANSWER: \((I+A)^{-1}=I-\frac{1}{1+\|\vec{v}\|^{2}} A\).]
in Mathematics by ♦ Platinum (93,241 points) | 319 views

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