Introduction to Matrix Diagonalization
- Diagonalization is a process used in linear algebra to simplify a matrix by converting it into a diagonal matrix.
- This process can make matrix computations much easier.
- It is an application of Eigenvalues and Eigenvectors
When is a Matrix Diagonalizable?
A square matrix A is diagonalizable if:
- It has n linearly independent eigenvectors (where n is the size of the matrix).
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Matrix Size:
The size of a square matrix A is the number of rows (or columns) it has.
For example:
A (2 × 2) matrix has size 2.
A 3 × 3 matrix has size 3.
What This Means for Diagonalization:
To diagonalize a matrix A, you need a total of n eigenvectors (where n is the size of the matrix) that are linearly independent.
If you can’t find enough independent eigenvectors, the matrix cannot be diagonalized.
For example:
If A is a 3 × 3 matrix, you need 3 independent eigenvectors.
If A is diagonalizable, we can write:
\[\LARGE
A = \color{red}{P} \color{blue}{D} \color{green}{P^{-1}}
\]
Which of the following statements about diagonalization is TRUE?