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alt="StartLayout 1st Row 1 period Matrix addition left-parenthesis and subtraction right-parenthesis period 2nd Row 2 period Scalar multiplication period 3rd Row 3 period Matrix multiplication period 4th Row 4 period Matrix transpose period 5th Row 5 period Trace left-parenthesis s u m of diagonal elements of normal a matrix right-parenthesis period EndLayout"/>

       Matrix addition:

       Scalar multiplication:

       Matrix multiplication:

       Transpose of a matrix AA new matrix obtained by writing the rows of A as columns.Applicable to rectangular matrices:

      1 Matrix trace (the sum of the diagonal element of a square matrix):

      An example of matrix multiplication is:

5.6 ESSENTIAL MATRIX TYPES

In this section we classify matrices based on some fundamental (computed) property. These properties are used in various areas of numerical analysis and its applications. It is then a useful reference to group them as we do here. In general, matrices can have real or complex values. In the latter case we recall complex conjugation:

      In many cases we are concerned with square matrices with real values.

      5.6.1 Nilpotent and Related Matrices

      A nilpotent matrix A is a square matrix such that for some positive integer p. It is said to be of index p if p the least positive integer for which . For example, the matrix:

      is nilpotent with index 2. More generally, a triangular matrix of size n with zeros along the main diagonal is nilpotent with index . For example, the follow matrix is nilpotent with index 3:

      The determinant and trace of a nilpotent matrix are always zero. Thus, such matrices are not invertible. However, and are invertible where N is a nilpotent matrix:

      (5.16)

      where I is the identity matrix. Since there are only finitely many non-zero terms, we see that both sums converge.

      An idempotent matrix A is one for which . Examples are:

      Idempotent matrices arise in regression analysis and econometrics, for example in ordinary least squares problems, in particular when estimating sums of squared residuals.

An involutory matrix is one that is its own inverse:

      An example is:

      For example, the Pauli matrices are involutory:

      5.6.2 Normal Matrices

      We introduce the important class of normal matrices by introducing some prerequisite notation. The transpose of a real matrix is an matrix formed by exchanging the rows and columns of A:

      In the case of a complex matrix A, the Hermitian transpose is the complex conjugate transpose of A:

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