# eigenvalue-eigenvector-eye-catch

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- Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix Consider the $2\times 2$ complex matrix \[A=\begin{bmatrix} a & b-a\\ 0& b \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenvectors. (c) Diagonalize the matrix $A$. (d) Using the result of the […]
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- Simple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
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- Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $? Let $A$ and $B$ be $n \times n$ matrices. Is it always true that $\tr (A B) = \tr (A) \tr (B) $? If it is true, prove it. If not, give a counterexample. Solution. There are many counterexamples. For one, take \[A = \begin{bmatrix} 1 & 0 \\ 0 & 0 […]
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