{"id":720,"date":"2016-01-29T23:30:42","date_gmt":"2016-01-30T04:30:42","guid":{"rendered":"http:\/\/underthehood.blogwyrm.com\/?p=720"},"modified":"2022-07-28T06:27:07","modified_gmt":"2022-07-28T10:27:07","slug":"matrices-as-vectors","status":"publish","type":"post","link":"https:\/\/underthehood.blogwyrm.com\/?p=720","title":{"rendered":"Matrices as Vectors"},"content":{"rendered":"

This week’s post will be a short little dive into some of the lesser-known back-waters of linear algebra. As discussed in other posts<\/a>, a vector space is a rather generic thing with applications to a wide variety of situations that often look quite different than the traditional length-and-direction or column-array pictures that tend to dot the landscape. <\/p>\n

One particularly interesting application, if for no other reason than it helps break the fixation that a vector must look like an arrow or a vertically stacked set of numbers is the application to the set of $$2\\times2$$ matrices generically denoted as<\/p>\n

\\[ M = \\left[ \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right] \\; ,\\]<\/p>\n

where the unspecified numbers $$\\left\\{a, b, c, d\\right\\}$$ can be complex.<\/p>\n

There are two bases that are nice to use. One is the obvious ‘natural’ basis spanned by vectors <\/p>\n

\\[ \\left| v_1 \\right \\rangle = \\left[ \\begin{array}{cc} 1 & 0 \\\\ 0 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

\\[ \\left| v_2 \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

\\[ \\left| v_3 \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 0 \\\\ 1 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

and <\/p>\n

\\[ \\left| v_4 \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 0 \\\\ 0 & 1 \\end{array} \\right] \\; .\\]<\/p>\n

The matrix $$M$$ is then decomposed, by inspection, to be<\/p>\n

\\[ \\left| M \\right \\rangle = a \\left| v_1 \\right \\rangle + b \\left| v_2 \\right \\rangle + c \\left| v_3 \\right \\rangle + d \\left| v_4 \\right \\rangle \\; .\\]<\/p>\n

Note the use of Dirac notation for both the basis vectors and the matrix $$M$$. The reason for this notational standard is that it will suggest how to get the components without using inspection, which will be the goal for computer-based decomposition for the second basis discussed below. <\/p>\n

The usual way perform decomposition is to provide a natural inner product – bra with ket – so that, for example, <\/p>\n

\\[ a = \\left \\langle w_1 \\right | \\left . M \\right \\rangle \\; .\\]<\/p>\n

So what is the rule for the bra $$\\left \\langle w_1 \\right |$$? It can’t be the usual complex-conjugate, transpose since the right-hand side of the previous equation is a $$2\\times2$$ matrix but the left-hand side is a scalar. Clearly, an added ingredient is needed. And, as will be shown below, that added ingredient is taking the trace.<\/p>\n

How then to establish this? Start by assuming the usual definition of the bra; that is let’s ignore the trace piece for the moment and define<\/p>\n

\\[ \\left \\langle w_i \\right | = \\left| v_i \\right \\rangle^{\\dagger} \\; i = 1,2,3,4 \\; .\\]<\/p>\n

The basic requirement to impose on the bra-ket relationship is <\/p>\n

\\[ \\left \\langle w_i \\right | \\left . v_j \\right \\rangle = \\delta_{ij} \\; . \\]<\/p>\n

There are 16 possible combinations, but one really only need look at a subset to infer the pattern. For convenience, take $$i=1$$ and let $$j=1,2,3,4$$. The four resulting products are:<\/p>\n

\\[ \\left \\langle w_1 \\right | \\left. v_1 \\right \\rangle = \\left[ \\begin{array}{cc} 1 & 0 \\\\ 0 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

\\[ \\left \\langle w_1 \\right | \\left. v_2 \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

\\[ \\left \\langle w_1 \\right | \\left. v_3 \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

and<\/p>\n

\\[ \\left \\langle w_1 \\right | \\left. v_4 \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \\end{array} \\right] \\; .\\]<\/p>\n

The other 12 products are similar. For each value of $$i$$, the matrix corresponding to $$j=i$$ has a single $$1$$ on the diagonal. Of the three matrices that result from $$j \\neq i$$, two matrices are identically zero and the other one has a $$1$$ on the off-diagonal. So to get a single outcome, $$0$$ for $$j \\neq i$$ and $$1$$ for $$j=i$$, simply take the trace of the resulting matrix.<\/p>\n

This algorithm applies equally well to the second basis, which is one that is based a set of matrices commonly found in modern physics. This basis is spanned by the $$2\\times2$$ identity matrix and the Pauli matrices:<\/p>\n

\\[ \\left| I \\right \\rangle = \\left[ \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right] \\; ,\\]<\/p>\n

\\[ \\left| \\sigma_x \\right \\rangle = \\left[ \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

\\[ \\left| \\sigma_y \\right \\rangle = \\left[ \\begin{array}{cc} 0 & -i \\\\ i & 0 \\end{array} \\right] \\; ,\\]<\/p>\n

and <\/p>\n

\\[ \\left| \\sigma_z \\right \\rangle = \\left[ \\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right] \\; .\\]<\/p>\n

There are two important points to raise here. First, the bra-forms of this basis are identical to the basis itself. Second, each Pauli matrix is trace-free. Together these two properties simplify things. To see this, start with the observation (which is well established in quantum mechanics texts) that the Pauli matrices obey the equation<\/p>\n

\\[ \\sigma_i \\sigma_j = i \\epsilon_{i j k} \\sigma_k + \\delta_{ij} I \\; .\\]<\/p>\n

Thus the trace-enabled inner product (bra-ket) adaptation that we are discussing becomes<\/p>\n

\\[ \\left \\langle \\sigma_i \\right | \\left . \\sigma_j \\right \\rangle = 2 \\delta_{i j} \\; , \\]<\/p>\n

where the first term on the right-hand side is zero since the Pauli matrices are traceless.<\/p>\n

There are two other types of combinations to consider. One an inner product where one member is the identity matrix and the other is a Pauli matrix. That inner product is zero again because the Pauli matrices are trace-free. The second one inner product of the identity matrix with itself, which is simply $$2$$. <\/p>\n

Decomposition by inspection really isn’t possible for a generic form of $$M$$, but a simple algorithm to perform the decomposition based on the analysis above is easy to implement. The code to perform these computations in wxMaxima is:<\/p>\n

\ncompose_matrix_2d(list) := (
\n block([id2,sigma_x,sigma_y,sigma_z,M],
\n id2 : matrix([1,0],[0,1]),
\n\t\t sigma_x : matrix([0,1],[1,0]),
\n\t\t sigma_y : matrix([0,-\n\t\t sigma_z : matrix([1,0],[0,-1]),<\/p>\n

\t\t M : map(ratsimp,list[1]*id2 + list[2]*sigma_x + list[3]*sigma_y + list[4]*sigma_z)
\n )
\n)$<\/p><\/div>\n \n

\ndecompose_matrix_2d(M) := (
\n block([id2,sigma_x,sigma_y,sigma_z,spec],
\n id2 : matrix([1,0],[0,1]),
\n\t\t sigma_x : matrix([0,1],[1,0]),
\n\t\t sigma_y : matrix([0,-\n\t\t sigma_z : matrix([1,0],[0,-1]),
\n\t\t spec : [0,0,0,0],
\n\t\t spec[1] : ratsimp(matrix_trace_2d( id2 . M )\/2),
\n\t\t spec[2] : ratsimp(matrix_trace_2d( sigma_x . M )\/2),
\n\t\t spec[3] : ratsimp(matrix_trace_2d( sigma_y . M )\/2),
\n\t\t spec[4] : ratsimp(matrix_trace_2d( sigma_z . M )\/2),
\n\t\t spec
\n )
\n)$<\/div>\n \n
\nmatrix_trace_2d(M) := (
\n block( result : M[1,1] + M[2,2] )
\n)$<\/div>\n \n

For a general matrix<\/p>\n

\\[ M = \\left[ \\begin{array}{cc} i & -3 + 4 i \\\\ 19 & 6 + 5 i \\end{array} \\right] \\; \\]<\/p>\n

the components are<\/p>\n

\\[ M \\doteq \\left[ \\begin{array}{c} 3 + 3 i \\\\ 8 + 2 i \\\\ -2 – 11 i \\\\ -3 – 2 i \\end{array} \\right] \\] <\/p>\n

a result that the reader is invited to confirm by performing the linear combination and which the reader is challenged to arrive at more easily than the method given here.<\/p>\n","protected":false},"excerpt":{"rendered":"

This week’s post will be a short little dive into some of the lesser-known back-waters of linear algebra. As discussed in other posts, a vector space is a rather generic… Read more ><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-720","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/720","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=720"}],"version-history":[{"count":2,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/720\/revisions"}],"predecessor-version":[{"id":723,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/720\/revisions\/723"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=720"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=720"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=720"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}