2. TRIANGULAR MATRICES A square matrix in which all the entries above the main diagonal are zero is called lower triangular, and a square matrix in which all the entries below the main diagonal are zero is called upper triangular. A matrix that is either upper triangular or lower triangular is called triangular. Example 2 a 11 a 12 a 13 a 14 0 a 22 a 23 a 24 0 0 a 33 a 34 0 0 0 a 44 A general 4×4 Upper triangular matrix a 11 0 0 0 a 21 a 22 0 0 a 31 a 32 a 33 0 a 41 a 42 a 43 a 44 A general 4×4 lower triangular matrix

3. The following are four useful characterizations of triangular materices. A square matrix A =[ a ij ] is upper triangular if and only if the i th row starts with at least i -1 zeros. A square matrix A =[ a ij ] is lower triangular if and only if the j th column starts with at least j -1 zeros. A square matrix A =[ a ij ] is upper triangular if and only if a ij =0 for i>j. A square matrix A =[ a ij ] is lower triangular if and only if a ij =0 for i<j.

4. PROPERTIES OF TRIANGULAR MATRICES Theorem 1.7.1 The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular. A triangular matrix is invertible if and only if its diagonal entries are all nonzero. The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular.

5. SYMMETRIC MATRICES A square matrix A is called symmetric if A=A. T

6. Example It is easy to recognize symmetric matrices by inspection: The entries on the main diagonal may be arbitrary, but “mirror images” of entries across the main diagonal must be equal 1 4 5 4 -3 0 5 0 7

7. Theorem1.7.2 If A and B are symmetric matrices with the same size, and if k is any scalar, then: a)A is symmetric. b)A+B and A-B are symmetric. c)kA is symmetric. T

8. In general, a symmetric matrix need not o be invertible; for example, a square zero matrix is symmetric, but not invertible. However, if a symmetric matrix is invertible, then that inverse is also symmetric.

9. Theorem 1.7.3. If A is an invertible symmetric matrix, then Aˉ¹ is symmetric.

10. Theorem 1.7.4. If A is an invertible matrix, then AA and A A are also invertible. TT