Lecture 12: Similar Matrices and Diagonalization | GO Classes Linear Algebra for GATE DA

Main Ideas

1. Similar Matrices

Two square matrices (A) and (B) are similar if there is an invertible matrix (S) such that [ B = S^{-1}AS. ] Equivalently, (A = SBS^{-1}). Similar matrices represent the same linear transformation in different bases. When the input and output use the same changed basis, the transformation’s matrix becomes (S^{-1}AS).

Similarity is an equivalence relation:

• Reflexive: Every matrix is similar to itself, using the identity matrix.
• Symmetric: If (A) is similar to (B), then (B) is similar to (A).
• Transitive: If (A) is similar to (B), and (B) is similar to (C), then (A) is similar to (C).

2. Properties Shared by Similar Matrices

Similar matrices share several important properties:

• Characteristic polynomial and eigenvalues: [ \det(B-\lambda I) = \det!\left(S^{-1}(A-\lambda I)S\right) = \det(A-\lambda I). ] Therefore, similar matrices have the same characteristic polynomial and eigenvalues.

• Determinant and trace: Similar matrices have the same determinant and trace. The determinant is also the product of the eigenvalues.

• Invertibility: One matrix is invertible if and only if the other is.
• Rank and nullity: Similarity preserves rank. The invertible change of coordinates maps one null space bijectively to the other, so their dimensions—and therefore the ranks—match.

Eigenvectors themselves are generally not identical for similar matrices; they are related by the change-of-basis matrix. A mismatch in an invariant such as trace can show that two matrices are not similar. However, matching invariants alone does not generally prove that matrices are similar.

3. Diagonal Matrices

A matrix is diagonal if every entry outside the main diagonal is zero. Its diagonal entries may be zero or nonzero; the zero matrix is also diagonal.

4. Diagonalization

A matrix (A) is diagonalizable if it is similar to a diagonal matrix. To construct a diagonalization:

1. Find the eigenvalues of (A).
2. Find corresponding eigenvectors.
3. Place (n) linearly independent eigenvectors in the columns of a matrix (V).
4. Place the corresponding eigenvalues on the diagonal of (\Lambda), in the same order.
5. Then [ AV = V\Lambda. ] If (V) is invertible, this gives [ A = V\Lambda V^{-1}. ]

Thus, an (n \times n) matrix is diagonalizable exactly when it has (n) linearly independent eigenvectors.

Key points:

• (n) distinct eigenvalues guarantee (n) linearly independent eigenvectors, so the matrix is diagonalizable.
• Repeated eigenvalues do not automatically prevent diagonalization; the matrix may still have enough linearly independent eigenvectors.
• A real symmetric matrix has orthogonal eigenvectors. After normalizing them, the eigenvector matrix (Q) is orthogonal, so (Q^{-1} = Q^T).

5. Why Diagonalization Is Useful

If (A = PDP^{-1}), then [ A^k = PD^kP^{-1}. ] Taking powers of a diagonal matrix is straightforward: raise each diagonal entry to the (k)th power. This makes diagonalization useful for computing matrix powers.

Diagonalizability and invertibility are separate properties; neither guarantees the other. For example, a diagonal matrix with a zero on its diagonal is diagonalizable but not invertible, while an invertible matrix need not be diagonalizable.

6. Practice and Upcoming Topics

The instructor assigns matrix-diagonalization problems as homework for discussion in the next lecture. The announced upcoming topic is singular value decomposition (SVD), followed later by principal component analysis (PCA).

Speakers and Sources

• Instructor: Unnamed lecturer from GO Classes for GATE DA.
• Audience or chat participants named in the subtitles: Raj, Sonam, Samir, Arjun, Rakesh, S. Sahu, and H. The subtitles do not establish whether each contributed an audible comment or a chat message.
• Other source mentioned: The instructor says the diagonalization theorem was taken from Berkeley University.