1. Tutorial 3.

2. 2 Gauss-Jordan elimination. 1. Calculate the inverse matrix.

3. 3 Inverse Matrix

4. 4 Consider the subset of indexes Then the matrix is called a leading minor of A Leading minors

5. 5 2. Prove: The leading minors of positive (/semi) definite matrix are positive (/semi) definite. Assume that the statement is false. Then,

6. 6 Consider the vector :

7. 7 We obtained, contradiction to the assumption that therefore Corollary: Diagonal elements of positive definite (semi-definite) matrix A are positive (nonnegative). Proof: In the proof for general case, we now take and obtain

8. 8 Consider direct multiplication requires multiplications. The naive partitioning doesn’t help: 3. Fast Matrix Multiplication (Strassen‘69)

9. 9 Consider then

10. 10 Number of operations More precisely, Practically, the gain is not so large: