Find an eigenbasis for each of the matrices A in Exercises...

Find an eigenbasis for each of the matrices $A$ in Exercises 50 through $54,$ and thus diagonalize A. Hint: Exercise 48 is helpful.
$$A=\left[\begin{array}{rrr}
1 & -1 & 1 \\
-1 & 1 & -1 \\
1 & -1 & 1
\end{array}\right]$$

Key Concepts

Multiplicity (Algebraic and Geometric)

Algebraic multiplicity refers to the number of times an eigenvalue appears as a root of the characteristic polynomial, while geometric multiplicity is the number of independent eigenvectors associated with that eigenvalue. For a matrix to be diagonalizable, the geometric multiplicity must equal the algebraic multiplicity for each eigenvalue.

Diagonalization

Diagonalization is the process of finding a diagonal matrix D and an invertible matrix P such that A = PDP?¹, where the columns of P are the eigenvectors of A and the diagonal entries of D are the corresponding eigenvalues. This process simplifies matrix computations and helps in understanding the structure of the linear transformation.

Characteristic Polynomial

The characteristic polynomial of a matrix is obtained by calculating det(A - ?I), and its roots are the eigenvalues of the matrix. This polynomial encapsulates the key information about the eigenstructure of the matrix.

Eigenvectors

Eigenvectors are the nonzero vectors v that satisfy the equation Av = ?v for a given eigenvalue ?. They form the building blocks for constructing an eigenbasis, and each eigenvector shows the directions in which the linear transformation acts by simply scaling.

Eigenvalues

Eigenvalues are scalars ? that satisfy the equation Av = ?v for a nonzero vector v. They are found by solving the characteristic equation det(A - ?I) = 0. In the context of diagonalization, eigenvalues are the entries on the diagonal of the diagonal matrix to which the original matrix is similar.

Eigenbasis

An eigenbasis is a basis for the vector space that consists entirely of eigenvectors of the matrix. Having an eigenbasis is crucial for diagonalization because it means the matrix can be expressed in a basis where it acts as a simple scaling transformation.

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