Matrices - Operations with Matrices

Kamis, 07 Mei 2020

Operations with Matrices

• Two matrix A and B are equal if, and only if, they are both of the same shape m × n and corresponding elements are equal.

• Two matrix A and B can be added (or subtracted) of, and only if, they have the same shape m × n . If

     

• If k is a scalar, and A = ⌊ a_ij ⌋ is a matrix, then

     

• Multiplication of Two Matrices

Two matrices can be multiplied together only when the number of column in the first is equal to the number of rows in the second If

    

• Transpose of a Matrix

If the rows and columns of a matrix are interchanged, then the new matrix is called the transpose of the original matrix. If A is the original matrix, its transpose is denoted A^T or  Ã.

•  The matrix A is orthogonal if AA^T = I

• If the matrix product AB is defined, then (AB)^T = B^T A^T.

• Adjoint of Matrix

If A is a square n × n matrix, its adjoint, denated by adj A, is the transpose of the matrix of cofactors C_ij of A: adj A = [C_ij]^T.

• Trace of a Matrix

If A is a square n × n matrix, its trace, denated by tr A, is defined to be the sum of the terms on the leading diagonal: tr A = a₁₁ + a₂₂ + … + a_nn.

• Inverse of a Matrix

If A is a square n × n matrix with a nonsingular determinant det A, then its inverse A^{– 1} is given by

    

• If the matrix product AB is defined, then (AB)^{– 1}  = B ^{– 1} A ^{– 1} .

• If A is a square n × n matrix, the eigenvectors X satisfy the equation

AX = λX,

While the eigenvalues λ satisfy the characteristic equation | A – λ I | = 0.

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Labels: Mathematician

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