Introductory Matrix Operations Matrix Entries Defn. For matrix A , - PowerPoint PPT Presentation

Introductory Matrix Operations

Matrix Entries Defn. For matrix A , notation a ij means the en- try in row i and column j of A . matOpsONE: 2

Matrix Addition and Scalar Multiplication Matrix addition requires the two ma- Defn. trices have the same dimensions. The sum is defined by adding corresponding entries. Sim- ilarly, scalar multiplication is defined entry- wise. For example, � a 11 a 12 a 13 � � b 11 b 12 b 13 � � a 11 + b 11 a 12 + a 12 a 13 + b 13 � + = a 21 + b 21 a 22 + a 22 a 23 + b 23 a 21 a 22 a 23 b 21 b 22 b 23 � a 11 a 12 � � ca 11 ca 12 � and = c a 21 a 22 ca 21 ca 22 matOpsONE: 3

Matrix Transpose Defn. The transpose of a matrix A , denoted A T , exchanges rows and columns. That is, ( A T ) ij = A ji . For example: here is a matrix and its transpose   � 3 4 7 3 − 2 � 4 5   − 2 5 − 3   7 − 3 matOpsONE: 4

Square Matrices Defn. A square matrix has equal number of rows and columns. matOpsONE: 5

Diagonal Matrices Defn. The diagonal of a square matrix runs from top-left to bottom-right. A diagonal ma- trix is a square matrix that has zeros off the diagonal (and might or might not have zeroes on the diagonal). For example   3 0 0 0 − 1 0     0 0 − π matOpsONE: 6

Symmetric Matrices Defn. A symmetric matrix is a square matrix that is symmetric around its diagonal. In other words, A = A T . matOpsONE: 7

Summary Matrix addition and scalar multiplication is de- fined entry-wise. For matrix A , notation a ij means the entry in row i and column j of A . The transpose of a matrix exchanges rows and columns. A square matrix has equal number of rows and columns. The diagonal of a square matrix runs from top-left to bottom-right. A di- agonal matrix has zeros off the diagonal. A sym- metric matrix equals its transpose. matOpsONE: 8