Section 1.7 - Special Matrices A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix . For any diagonal matrix D we must have dij : 0 if i é j. Examples: 1 0 0 0 5 0 0 0 0 0 9 0 0 1 0 0 0 ?2 0 0 0 0 0 ?6 0 0 1 0 0 0 0 0 0 0 0 0 0 1 A general n ≥ n diagonal matrix can be written as d1 0 0 0 d2 0 D : _ _ b _ 0 0 dn Inverses , Powers ,& Products of Diagonal Matrices : Diagonal matrices behave very much like real numbers when we look at inverses, powers, & products. An n ≥ n diagonal matrix D is invertible if and only if all entries along the main diagonal are NONZERO . If D is invertible, then ?1 d 1 0 0 0 d ?1 0 D?1 : 2 _ _ b _ ?1 0 0 d n 3 0 1 0 For example, if D : , then D?1 : 3 . Confirm that DD ?1 : D?1D : I 0 4 5 5 0 4 1 5 0 Consider D : and compute D2 0 2 In general, for any positive integer k: k d1 0 0 0 d k 0 Dk : 2 _ _ b _ k 0 0 dn Consider the following product: 4 0 2 0 : 0 3 0 5 In general, d1 0 0 e1 0 0 d1e1 0 0 0 d2 0 0 e2 0 0 d2e2 0 : _ _ b _ _ _ b _ _ _ b _ 0 0 dn 0 0 en 0 0 dnen Also, If A and B are both n ≥ n diagonal matrices, then AB : BA . 2 A square matrix in which all the entries ABOVE the main diagonal are zero is called lower triangular . A square matrix in which all the entries BELOW the main diagonal are zero is called upper triangular . a11 0 0 a11 a12 a13 m i 9 j L : a21 a22 0 U : 0 a22 a23 m a31 a32 a33 0 0 a33 i ; j m If A : aij then we can classify triangular matrices as follows: Diagonal UpperTriangular LowerTriangular aij : 0 if i é j aij : 0 if i ; j aij : 0 if i 9 j Note that diagonal matrices are BOTH upper and lower triangular. The transpose of a lower triangular matrix is upper triangular, and the transpose of a 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 (iff) its main diagonal entries are nonzero. The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular. 2 0 0 ?3 0 0 Let A : ?1 5 0 and B : 1 0 0 0 2 1 ?9 4 8 Which of these matrices is invertible? Why? What can be said about the products AB and BA ? 3 A square matrix A is said to be symmetric if A : AT. A : aij is symmetric if aij : aji 3 4 ?1 Example: A : 4 ?2 0 ?1 0 8 Theorem : If A and B are both n ≥ n symmetric matrices and k is any scalar, then 1. AT is symmetric 2. A ? B and A + B are symmetric 3. kA is symmetric Proof : Note: If A and B are both n ≥ n symmetric matrices, then AB is NOT necessarily symmetric. This is true since AB T : BTAT : BA é AB If A is symmetric and invertible, then A?1 is symmetric since A?1 T : AT ?1 : A?1. A square matrix A is skew -symmetric if AT : ?A. Challenge : find a 3 ≥ 3 nonzero skew-symmetric matrix. 4.