Diagonalize Matrix Calculator

Eigenvalue 1 (lambda 1)Diagonalizable

First eigenvalue of matrix A

Eigenvalue 2 (lambda 2)3

P matrix (eigenvectors as columns)[[1, -0.7071], [0, 0.7071]]

D matrix (diagonal eigenvalues)[[4, 0], [0, 3]]

P-inverse matrix[[1, 1], [0, 1.4142]]

Verification error (max |A - PDP-1|)0

DiagonalizabilityDiagonalizable. A = P D P-1 verified.

Eigenvalue 14

Eigenvalue 23

Eigenvalue 3-

Your 2x2 matrix is diagonalizable.

The eigenvalues are 4 and 3. Each is a scalar by which its eigenvector is stretched (or flipped) when multiplied by A.
All eigenvalues are distinct, which guarantees diagonalizability: distinct eigenvalues always produce linearly independent eigenvectors.
Verification: the maximum entry-wise error in A vs. PDP-1 is 0.00e+0, confirming the factorization is numerically accurate.
Use the D matrix to compute matrix powers easily: A^k = P D^k P-1, and D^k is found by raising each diagonal entry to the k-th power.

Next stepTo compute A^k for any positive integer k, replace each diagonal entry of D with its k-th power, then multiply P D^k P-1.

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