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

Introductory Matrix Operations

Matrix Entries

Defn. For matrix A, notation aij means the en- try in row i and column j of A.

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Matrix Addition and Scalar Multiplication

Defn. Matrix addition requires the two ma- trices have the same dimensions. The sum is defined by adding corresponding entries. Sim- ilarly, scalar multiplication is defined entry- wise. For example, a11 a12 a13 a21 a22 a23

• +

b11 b12 b13 b21 b22 b23

• =

a11 + b11 a12 + a12 a13 + b13 a21 + b21 a22 + a22 a23 + b23

• and

c a11 a12 a21 a22

• =

ca11 ca12 ca21 ca22

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Matrix Transpose

Defn. The transpose of a matrix A, denoted AT, exchanges rows and columns. That is, (AT)ij = Aji. For example: here is a matrix and its transpose 3 4 7 −2 5 −3

• 

  3 −2 4 5 7 −3   

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Square Matrices

Defn. A square matrix has equal number of rows and columns.

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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

• n the diagonal).

For example    3 0 0 −1 0 0 −π   

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Symmetric Matrices

• Defn. A symmetric matrix is a square matrix

that is symmetric around its diagonal. In other words, A = AT.

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Summary

Matrix addition and scalar multiplication is de- fined entry-wise. For matrix A, notation aij 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

• f 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.

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