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HARVARD MATH 21B - MATRIX PRODUCT

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MATRIX PRODUCT Math 21b, O. KnillHOMEWORK: Section 2.4: 4,14,28,40,76,48*,66*MATRIX PRODUCT. If B is a p ×m matrix and A is a m ×n matrix, then BA is defined as the p ×n matrixwith entries (BA)ij=Pmk=1BikAkj.EXAMPLE. If B is a 3 × 4 matrix, and A is a 4 × 2 matrix then BA is a 3 × 2 matrix.B =1 3 5 73 1 8 11 0 9 2, A =1 33 11 00 1, BA =1 3 5 73 1 8 11 0 9 21 33 11 00 1=15 1314 1110 5.COMPOSING LINEAR TRANSFORMATIONS. If S : Rn→ Rm, x 7→ Ax and T : Rm→ Rp, y 7→ By arelinear transformations, then their composition T ◦ S : x 7→ BA(x)) is a linear transformation from Rnto Rp.The corresponding matrix is the matrix product B · A.EXAMPLE. Find the matrix which is a composition of a rotation around the x-axes by an agle π/2 followedby a rotation around the z-axes by an angle π/2.SOLUTION. The first transformation has the property that e1→ e1, e2→ e3, e3→ −e2, the second e1→e2, e2→ −e1, e3→ e3. If A is the matrix belonging to the first transformation and B the second, then BAis the matrix to the composition. The composition maps e1→ −e2→ e3→ e1is a rotation around a longdiagonal. B =0 −1 01 0 00 0 1A =1 0 00 0 −10 1 0, BA =0 0 11 0 00 1 0.EXAMPLE. A rotation dilation is the composition of a rotation by α = arctan(b/a) and a dilation (=scale) byr =√a2+ b2.REMARK. Matrix multiplication is a generalization of usual multiplication of numbers or the dot product.MATRIX ALGEBRA. Note that AB 6= BA in general! Otherwise, the same rules apply as for numbers:A(BC) = (AB)C, AA−1= A−1A = 1n, (AB)−1= B−1A−1, A(B + C) = AB + AC, (B + C)A = BA + CAetc.PARTITIONED MATRICES. The entries of matrices can themselves be matrices. If B is a m × n matrix andA is a n × p matrix, and assume the entries are k × k matrices, then BA is a m × p matrix where each entry(BA)ij=Pnk=1BikAkjis a k ×k matrix. Partitioning matrices can be useful to improve the speed of matrixmultiplication (i.e. Strassen algorithm).EXAMPLE. If A =A11A120 A22, where Aijare k × k matrices with the property that A11and A22areinvertible, then B =A−111−A−111A12A−1220 A−122is the inverse of A.APPLICATIONS. (The material which follows is for motivation puposes only, more applications appear in thehomework).1423NETWORKS. Let us associate to the computer network (shown at the left) amatrix0 1 1 11 0 1 01 1 0 11 0 1 0To a worm in the first computer we associate a vector1000. The vector Ax has a 1 at the places, where the worm could be in the nextstep. The vector (AA)(x) tells, in how many ways the worm can go from the firstcomputer to other hosts in 2 steps. In our case, it can go in three different waysback to the computer itself.Matrices help to solve combinatorial problems (see movie ”Good will hunting”).For example, what does [A1000]22tell about the worm infection of the network?What does it mean if A100has no zero entries?FRACTALS. Closely related to linear maps are affine maps x 7→ Ax + b. They are compositions of a linearmap with a translation. It is not a linear map if B(0) 6= 0. Affine maps can be disguised as linear mapsin the following way: let y =x1and defne the (n+1)∗(n+1) matrix B =A b0 1. Then By =Ax + b1.Fractals can be constructed by taking for example 3 affine maps R, S, T which contract area. For a given objectY0define Y1= R(Y0) ∪ S(Y0) ∪ T (Y0) and recursively Yk= R(Yk−1) ∪ S(Yk−1) ∪ T (Yk−1). The above pictureshows Ykafter some iterations. In the limit, for example if R(Y0), S(Y0) and T (Y0) are disjoint, the sets Ykconverge to a fractal, an object with dimension strictly between 1 and 2.CHAOS. Consider a map in the plane like T :xy7→2x + 2 sin(x) − yxWe apply this map again andagain and follow the points (x1, y1) = T (x, y), (x2, y2) = T (T (x, y)), etc. One writes Tnfor the n-th iterationof the map and (xn, yn) for the image of (x, y) under the map Tn. The linear approximation of the map at apoint (x, y) is the matrix DT (x, y) =2 + 2 cos(x) − 11. (If Txy=f(x, y)g(x, y), then the row vectors ofDT (x, y) are just the gradients of f and g). T is called chaotic at (x, y), if the entries of D(Tn)(x, y) growexponentially fast with n. By the chain rule, D(Tn) is the product of matrices DT (xi, yi). For example, T ischaotic at (0, 0). If there is a positive probability to hit a chaotic point, then T is called chaotic.FALSE COLORS. Any color can be represented as a vector (r, g, b), where r ∈ [0, 1] is the red g ∈ [0, 1] is thegreen and b ∈ [0, 1] is the blue component. Changing colors in a picture means applying a transformation on thecube. Let T : (r, g, b) 7→ (g, b, r) and S : (r, g, b) 7→ (r, g, 0). What is the composition of these two linear maps?OPTICS. Matrices help to calculate the motion of light rays through lenses. Alight ray y(s) = x + ms in the plane is described by a vector (x, m). Followingthe light ray over a distance of length L corresponds to the map (x, m) 7→(x + mL, m). In the lens, the ray is bent depending on the height x. Thetransformation in the lens is (x, m) 7→ (x, m − kx), where k is the strength ofthe lense.xm7→ ALxm=1 L0 1xm,xm7→ Bkxm=1 0−k 1xm.Examples:1) Eye looking far: ARBk. 2) Eye looking at distance L: ARBkAL.3) Telescope: Bk2ALBk1. (More about it in problem 80 in section


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HARVARD MATH 21B - MATRIX PRODUCT

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