The diagonalizable case was discussed in the other problem and gives a superset of the examples I gave. When the two matrices are simultaneously diagonalizable then the matrices commute. The examples in the list above are in fact valid even when the matrices are not diagonalizable. Sign up to join this community. The best answers are voted up and rise to the top.
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Learn more. Asked 3 years, 7 months ago. Active 3 years, 7 months ago. Viewed 46k times. Add a comment. Active Oldest Votes. Since two matrices are equal if and only if they are of the same size and their corresponding entries are equal, this last equation implies.
Scalar multiplication. A matrix can be multiplied by a scalar as follows. That is, the matrix kA is obtained by multiplying each entry of A by k. This definition of matrix subtraction is consistent with the definition illustrated in Example 8. Matrix multiplication. By far the most important operation involving matrices is matrix multiplication , the process of multiplying one matrix by another.
The first step in defining matrix multiplication is to recall the definition of the dot product of two vectors. Writing r as a 1 x n row matrix and c as an n x 1 column matrix, the dot product of r and c is. Note that in order for the dot product of r and c to be defined, both must contain the same number of entries.
Also, the order in which these matrices are written in this product is important here: The row vector comes first, the column vector second. Now, for the final step: How are two general matrices multiplied?
First, in order to form the product AB, the number of columns of A must match the number of rows of B ; if this condition does not hold, then the product AB is not defined. This criterion follows from the restriction stated above for multiplying a row matrix r by a column matrix c , namely that the number of entries in r must match the number of entries in c. The following diagram is helpful in determining if a matrix product is defined, and if so, the dimensions of the product:.
Thinking of the m x n matrix A as composed of the row vectors r 1 , r 2 ,…, r m from R n and the n x p matrix B as composed of the column vectors c 1 , c 2 ,…, c p from R n ,. Since A is 2 x 3 and B is 3 x 4, the product AB , in that order, is defined, and the size of the product matrix AB will be 2 x 4. The product BA is not defined, since the first factor B has 4 columns but the second factor A has only 2 rows.
The number of columns of the first matrix must match the number of rows of the second matrix in order for their product to be defined. Taking the dot product of row 1 in A and column 1 in B gives the 1, 1 entry in AB. The dot product of row 1 in A and column 2 in B gives the 1, 2 entry in AB ,.
The first row of the product is completed by taking the dot product of row 1 in A and column 4 in B , which gives the 1, 4 entry in AB :. Finally, taking the dot product of row 2 in A with columns 3 and 4 in B gives respectively the 2, 3 and 2, 4 entries in AB :. First, note that since C is 4 x 5 and D is 5 x 6, the product CD is indeed defined, and its size is 4 x 6. The 3, 5 entry of CD is the dot product of row 3 in C and column 5 in D :. The previous example gives one illustration of what is perhaps the most important distinction between the multiplication of scalars and the multiplication of matrices.
However, it is decidedly false that matrix multiplication is commutative. In fact, the matrix AB was 2 x 2, while the matrix BA was 3 x 3. Here is another illustration of the noncommutativity of matrix multiplication: Consider the matrices.
Since C is 3 x 2 and D is 2 x 2, the product CD is defined, its size is 3 x 2, and. The product DC , however, is not defined, since the number of columns of D which is 2 does not equal the number of rows of C which is 3. Since A is 2 x 2, in order to multiply A on the right by a matrix, that matrix must have 2 rows.
Therefore, if x is written as the 2 x 1 column matrix. Show that any two square diagonal matrices of order 2 commute. Although matrix multiplication is usually not commutative, it is sometimes commutative; for example, if. Despite examples such as these, it must be stated that in general, matrix multiplication is not commutative. There is another difference between the multiplication of scalars and the multiplication of matrices. That is, the only way a product of real numbers can equal 0 is if at least one of the factors is itself 0.
The analogous statement for matrices, however, is not true. For instance, if. Note that even though neither G nor H is a zero matrix, the product GH is. Yet another difference between the multiplication of scalars and the multiplication of matrices is the lack of a general cancellation law for matrix multiplication. For example, if. Example 13 : Although matrix multiplication is not always commutative, it is always associative. That is, as long as the order of the factors is unchanged, how they are grouped is irrelevant.
Note that the associative law implies that the product of A, B , and C in that order can be written simply as ABC ; parentheses are not needed to resolve any ambiguity, because there is no ambiguity. In fact, the equation. This says that if the product AB is defined, then the transpose of the product is equal to the product of the transposes in the reverse order.
Identity matrices. The zero matrix 0 m x n plays the role of the additive identity in the set of m x n matrices in the same way that the number 0 does in the set of real numbers recall Example 7. Similarly, the matrix.
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