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Identity matrix - Wikipedia
Nondefault Numeric Data Type. Complex Identity Matrix. Sparse Identity Matrix. Input Arguments collapse all n — Size of first dimension of I integer value. Size of first dimension of I , specified as an integer value. Size of second dimension of I , specified as an integer value. Prototype, specified as a numeric variable. Size arguments must have a fixed size. Using two iterators, locate each position of matrix and check whether the right diagonal have all 1s with remaining elements as 0s. Set a flag variable 1 if it matches the above condition, otherwise it will be 0.
Print Identity matrix if flag is 1. Here is source code of the C program to check a given Matrix is an Identity Matrix. The program is successfully compiled and tested using Turbo C compiler in windows environment.
Properties of matrix multiplication
The program output is also shown below. It follows straightforwardly from proposition 2.
Indeed, taking in 2. If P and Q are nonsingular and A is square, then. Read full chapter. Therefore the ith column of D is identical to the j th column of C , and thus their interchange, by Theorem , will not affect C. This proves the corollary. The trace of A will, sometimes, be denoted by Tr A or tr A. Vladimir Britanak, Rao, in Discrete Cosine and Sine Transforms , In particular, the block matrix of the form:.
Generally, the algebra of square quasi-triangular block matrices such as addition and multiplication is similar to that of square triangular matrices with scalar elements. In particular  :.
The product of two lower upper quasi-triangular matrices is lower upper quasi-triangular matrix. The determinant of quasi-triangular matrix and quasi-diagonal matrix too with square diagonal blocks is equal to the product of determinants of diagonal square block matrices. The transposition of lower upper quasi-triangular matrix is upper lower quasi-triangular matrix.
Obviously, the inverses of quasi-triangular matrices given by 5. The factorization 5. It can be extended to cases where each scalar element in the factored matrices is a square nonsingular matrix. In fact, for an arbitrary invertible matrix T N of order N the following factorization is possible [ 68—70 ]:.
An alternative factorization of a quasi-diagonal matrix with opposite main diagonal is defined as . A reversible integer-to-integer mapping is simply constructed by the implementation of matrix—vector multiplications in floating-point arithmetic with the results rounded to the nearest integer.
If the integer input vector is applied to a matrix on the right-hand side of 5. Hence, the large parts of the transforms are computed without rounding operations, only the results of matrix-vector multiplications are rounded and added. A different factorization of the general block transform matrix into the product of lower and upper quasi-triangular matrices and quasi-diagonal matrix has been proposed in Refs.
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If the factorization 5. The MDL computational structure defined by 5. Moreover, substituting the factorization 5. In the construction of integer DCTs, frequently the floating-point multipliers are either approximated by dyadic rationals or are rounded to the nearest integer.
Both dyadic approximation and the rounding procedure introduce truncation errors. In Section 5. On the other hand, in Section 5. However, a little attention has been paid to the analysis of errors caused by dyadic approximation. Moreover, the explicit estimates of truncation errors have been derived as follows. For simplicity, let us use the notation used in Section 5.
Theorem 5. Further, the componentwise truncation error can be estimated by.
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