Diagonal Matrix Multiplication Properties

13 Apr, 2021

A diagonal matrix is a square matrix in which all entries are zero except for those on the leading diagonal. A vector space is a set equipped with two operations vector addition and scalar multiplication satisfying certain properties.

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Lets learn about the properties of the diagonal matrix now.

Diagonal matrix multiplication properties. Deﬁnition The transpose of an m x n matrix A is the n x m matrix AT obtained by interchanging rows and columns of A Deﬁnition A square matrix A is symmetric if AT A. Matrix multiplication The product of matrices AandBis deﬁned if thenumber of columns inAmatches the number ofrows inB. A diagonal matrix is a square matrix whose entries outside of the main diagonal are all zero.

3 If B is a diagonal matrix then B T is a diagonal matrix. A special case of a symmetric matrix is a diagonal matrix. Proof A diagonal matrix is triangular and a triangular matrix is invertible if and only if all the entries on its main diagonal are non-zero.

C ii a ii b ii and all other entries are 0. If A and B are diagonal then C AB BA. 2 If B is a diagonal matrix then cB is a diagonal matrix for any scalar c.

It is also called the scaling matrix because multiplication with the diagonal matrix scales an object in a corresponding vector space. If addition or multiplication is being applied on diagonal matrices then the matrices should be of the same order. Matrix transpose AT 15 33 52 21 A 1352 532 1 Example Transpose operation can be viewed as ﬂipping entries about the diagonal.

The above identity simplifies to One can check that this last condition implies must be a diagonal matrix in other words the only triangular matrix which is also normal is a diagonal. Further C can be computed more efficiently than naively doing a full matrix multiplication. ABC ABC associative law ABC AC BC distributive law 1 CAB CACB distributive law 2 rAB ArB rAB Any of the above identities holds provided that matrix sums and products are well deﬁned.

D D T. TheproductABisdeﬁned to be thempmatrixC cij such thatcijPnaikbkj for. P Q.

21 Matrix Multiplication Properties 1 A square zero matrix is a diagonal matrix. Properties of matrix multiplication. P Q.

Properties of Diagonal Matrix In this section you will be studying the properties of the diagonal matrix. Transpose of the diagonal matrix D is as the same matrix. We introduce matrices define matrix addition and scalar multiplication and prove properties of those operations.

Multiplication of diagonal matrices is commutative. Same order diagonal matrices gives a diagonal matrix only after addition or multiplication. An example of a diagonal matrix The following matrices may be confused as diagonal matrices.

If A and B are diagonal then C AB is diagonal. 4 if B and C are diagonal matrices of the same size then B C is a diagonal matrix. Diagonal matrices have some properties that can be usefully exploited.

LetA aik be anmnmatrix and bkj be annpmatrix. Ie AT ij A ji ij. Proposition A diagonal matrix is invertible if and only if all the entries on its main diagonal are non-zero.