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Matrix Operations - Deep Dive

Mathematical Foundations​

Rigorous treatment of matrix operations, factorizations, and special matrix properties.

Matrix Multiplication Theory​

Algebraic Properties​

Theorem: Matrix multiplication is associative but not commutative.

Proof: (To be completed)

Block Multiplication​

(Mathematical treatment to be completed)

LU Decomposition​

Existence Theorem​

Theorem: An n×n matrix A has an LU decomposition if and only if...

Proof: (To be completed)

Algorithm​

(Detailed algorithm to be completed)

Special Matrices​

Symmetric Matrices​

Theorem: If A and B are symmetric, then...

Proof: (To be completed)

Orthogonal Matrices​

Theorem: Q is orthogonal if and only if QᵀQ = I

Properties: (To be completed)

Computational Complexity​

Analysis​

(To be completed)

Exercises​

(Advanced problems to be completed)