Can someone show me step-by-step how to diagonalize this matrix?

rheisf 2021-12-18 Answered
Can someone show me step-by-step how to diagonalize this matrix? Im
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William Appel
Answered 2021-12-19 Author has 44 answers
First step: Find the eigenvalues of your matrix.
Eigenvectors are vectors x such that upon being multiplied by a matrix A, they are only scaled by a number. That is Ax=λx, where λ is just a number, called the eigenvalue associated with the eigenvector x.
The way to do this is to subtract the λx from both sides to get Axλx=0. Now factor out the x to get (AλI)x=0, where I is the identity matrix - note: we need the identity matrix because adding a matrix and a scalar is undefined.
This equation, (AλI)x=0 has a nontrivial solution (a solution where x0) if and only if det(AλI)=0 (can you prove this?).
So lets

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lenkiklisg7
Answered 2021-12-20 Author has 29 answers
This equation, (AλI)x=0 has a nontrivial solution (a solution where x0) if and only if det(AλI)=0"
We first assume that B=AλI is invertible (or det(B)0). If this is true, the equation can be rewritten as:
B1Bx=B10=0
Since B1B=I (the identify matrix), then x=0, which is false. Then B is not invertible or det(B)=0.

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