Matrix Multiplication Notes Pdf

09 May, 2021

You can also multiply a matrix by a number by simply multiplying each entry of the matrix by the number. 1 2 5 6 3 4 7 8 15 27 16 28 35 47 36 48 P.

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Any matrix can be multiplied by a single number scalar.

Matrix multiplication notes pdf. The matrix B is the inverse of the matrix A and this is usually written as A1. A b c d. Aecf bedf agch bgdh.

In this chapter we will typically assume that our matrices contain only numbers. The resulting matrix will be m-by-k. Even if AB and BA are both deﬁned BA may not be the same size.

Multiplying Matrices Words Numbers Algebra In a matrix P AB each element pij is the sum of the products of consecutive entries in row i in matrix A and the column j in matrix B. The command for the identity matrix is eyen. The array 5 can be used re-.

Aebg af bh cedg cf dh e f g h. The kth power of a matrix A is the product of k copies of A. We next deﬁne the scalar multiple kX for a number k and a matrix X.

Equally the matrix A is the inverse of the matrix B. E f g h. A b c d.

Notes on Matrix Multiplication Since matrix multiplication is complicated we should expect that there are several ways to view it. Note that in order for the matrix product to exist the number of columns in A must equal the number of rows in B. 4 Matrix Multiplication We thought of Ax b as a combination of columns.

We just multiply every entry of X by k. 7 Multiplication by a number also satisﬁes the usual properties of number multiplication and a list. We will examine three approaches.

BA 3 4 3 1 2 2 7 11 9 4 3 2 5 6 3 3 5 2 1 0 0 0 1 0 0 0 1 I. Their diﬀerence is the matrix A Bdeﬁned by ABij aij bij. 10 2 015 The matrix consists of 6 entries or elements.

A matrix is basically an organized box or array of numbers or other expressions. U 32U0162O BKdu WtXae MSodfNtBwuafrKeE MLRLXCQH O QAjl PlF 1r siUg8h2t 4su crPeps9eHr0vOeld4. The same to find the entries in a matrix product.

Solution Using the rules of matrix multiplication AB 4 3 2 5 6 3 3 5 2 3 4 3 1 2 2 7 11 9 1 0 0 0 1 0 0 0 1 I. The product AB of two matrices A and B is deﬁned if and only if the number of columns of A is equal to the number of rows of B. Thus 8 2 6 3 7 16 48 24 56 Matrix multiplication involving a scalar is commutative.

There are many ways of looking at matrix multiplication and well start by examining a few special cases. We do this by multiplying all the entries of the matrix by that number. Matrix multiplication function consider function f.

We illustrate multiplication using two 2-by-2 matrices. Q v xMPad8eB Bwqi lt Ih n yIRnzf Ui3n WiSt teD VAdl9gxe Gbnr saX S2MK Worksheet by Kuta Software LLC. The first concerns the multiplication between a matrix and a scalar.

Matrix-vectorproduct very important special case of matrix multiplication. Direct Matrix multiplication of Given a matrix a matrix and a. Matrix multiplication is associative meaning that if A B and C are all n n matrices then ABC ABC.

Problems with hoping AB and BA are equal. Even if AB and BA are both deﬁned and of the same size they still may not be equal. There are several rules for matrix multiplication.

Any linear function f. Note that has entries and each entry takes time to compute so the total procedure takes time. 01 Thinking about entries of AB Recall that we dened the product AB of an m n matrix A with an n p matrix B by specifying how to compute each.

2 Matrix Multiplication The product of two matrices A Rmn and B Rnp is the matrix C AB Rmp where Cij Xn k1 AikBkj. A matrix Acan be multiplied by a scalar cto obtain the matrix cA where cAij caij. Y Ax A is an mn matrix x is an n-vector y is an m-vector y i A i1x1A inx n i 1m can think of y Ax as a function that transforms n-vectors into m-vectors a set of m linear equations relating x to y Matrix Operations 29.

That is for R aB then r ij ab ij for all i and j. Notes for the optimal splitting in computing. Speciﬁcally kX kx ij 1 i m1 j n For example For example 8 2 4 2 1 3 4 0 7 3 5 2 4 82 81 83 8 4 80 87 3 5 2 4 16 8 24 32 0 56 3 5.

However matrix multiplication is not commutative because in general AB 6 BA. Matrix operations are handled in two different fashions in Excel. Eg A is 2 x 3 matrix B is 3 x 5 matrix eg A is 2 x 3 matrix B is 3 x 2 matrix.

Addition of matrices and scalar multiplication are handled by conventional cell arithmetic copying cell formulas whereas advanced matrix operations such as transposition multiplication and inversion are handled by matrix. AB A 2 4 j j j b 1 b 2 b n j j j 3 5 2 4 j j j Ab 1 Ab 2 Ab n j j j 3 5 6. First the ﬁrst row of the left matrix is multiplied against and summed with the ﬁrst column of the right.

Here each element in the product matrix is simply the scalar multiplied by the element in the matrix. Evidently matrix multiplication is generally not commutative. Matrix multiplication not commutative In general AB BA.

Complexity of Direct Matrix multiplication. Rn Rm given by fx Ax where A Rmn matrix multiplication function f is linear converse is true. We can do the same thing with matrix multiplication.

Example Here is a matrix of size 2 3 2 by 3 because it has 2 rows and 3 columns. So if X x ij 1 i m1 j n is any m n matrix and k is any real number then kX is another m n matrix. If λ is a number and A is an nm matrix then we denote the result of such multiplication by λA where λA ij λA ij.

K a b c d ka kb kc kd For example 4 1 3 2 1 4 12 8 4 Matrix multiplication. This is called scalar multiplication. By entry by row and by column.

Rn Rm can be written as fx Ax for some A Rmn representation via matrix multiplication is unique. If the matrices Aand Bhave the same size then their sum is the matrix ABdeﬁned by ABij aij bij. Ak AAz A k times.

BA may not be well-deﬁned. Thats a little Matlab joke.

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