{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":3072,"title":"Singular Value Decomposition","description":"Calculate the three matrices of the singular value decomposition (A = U*S*V^T) for each provided matrix. U and V are square unitary matrices (V^T = the transpose of V) and S contains the singular values along the diagonal.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 21px; transform-origin: 407px 21px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCalculate the three matrices of the singular value decomposition (A = U*S*V^T) for each provided matrix. U and V are square unitary matrices (V^T = the transpose of V) and S contains the singular values along the diagonal.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [U,S,V] = sing_val_decomp(A)\r\n\r\n\r\nend","test_suite":"%%\r\nA = [3 0; 0 -2];\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(isequal(U,[1 0; 0 1]))\r\nassert(isequal(S,[3 0; 0 2]))\r\nassert(isequal(V,[1 0; 0 -1]))\r\n\r\n%%\r\nA = [0 2; 0 0; 0 0];\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(isequal(U,eye(3)))\r\nassert(isequal(S,fliplr(A)))\r\nassert(isequal(V,[0 -1; 1 0]))\r\n\r\n%%\r\nA = [3 1 1; -1 3 1];\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\nassert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\nassert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\r\n%%\r\nA = magic(3);\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\nassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\nassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tA = [3 0; 0 -2];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,[1 0; 0 1]))\r\n\t\tassert(isequal(S,[3 0; 0 2]))\r\n\t\tassert(isequal(V,[1 0; 0 -1]))\r\n\tcase 2\r\n\t\tA = [0 2; 0 0; 0 0];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,eye(3)))\r\n\t\tassert(isequal(S,fliplr(A)))\r\n\t\tassert(isequal(V,[0 -1; 1 0]))\r\n\tcase 3\r\n\t\tA = [3 1 1; -1 3 1];\r\n                [U,S,V] = sing_val_decomp(A);\r\n                assert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\n                assert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\n                assert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\tcase 4\r\n\t\tA = magic(3);\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tA = [3 0; 0 -2];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,[1 0; 0 1]))\r\n\t\tassert(isequal(S,[3 0; 0 2]))\r\n\t\tassert(isequal(V,[1 0; 0 -1]))\r\n\tcase 2\r\n\t\tA = [0 2; 0 0; 0 0];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,eye(3)))\r\n\t\tassert(isequal(S,fliplr(A)))\r\n\t\tassert(isequal(V,[0 -1; 1 0]))\r\n\tcase 3\r\n\t\tA = [3 1 1; -1 3 1];\r\n                [U,S,V] = sing_val_decomp(A);\r\n                assert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\n                assert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\n                assert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\tcase 4\r\n\t\tA = magic(3);\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tA = [3 0; 0 -2];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,[1 0; 0 1]))\r\n\t\tassert(isequal(S,[3 0; 0 2]))\r\n\t\tassert(isequal(V,[1 0; 0 -1]))\r\n\tcase 2\r\n\t\tA = [0 2; 0 0; 0 0];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,eye(3)))\r\n\t\tassert(isequal(S,fliplr(A)))\r\n\t\tassert(isequal(V,[0 -1; 1 0]))\r\n\tcase 3\r\n\t\tA = [3 1 1; -1 3 1];\r\n                [U,S,V] = sing_val_decomp(A);\r\n                assert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\n                assert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\n                assert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\tcase 4\r\n\t\tA = magic(3);\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\nend\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":4,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":37,"test_suite_updated_at":"2021-07-01T09:46:16.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-10T02:16:17.000Z","updated_at":"2026-03-19T18:40:04.000Z","published_at":"2015-03-10T02:16:17.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the three matrices of the singular value decomposition (A = U*S*V^T) for each provided matrix. U and V are square unitary matrices (V^T = the transpose of V) and S contains the singular values along the diagonal.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":3072,"title":"Singular Value Decomposition","description":"Calculate the three matrices of the singular value decomposition (A = U*S*V^T) for each provided matrix. U and V are square unitary matrices (V^T = the transpose of V) and S contains the singular values along the diagonal.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 21px; transform-origin: 407px 21px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCalculate the three matrices of the singular value decomposition (A = U*S*V^T) for each provided matrix. U and V are square unitary matrices (V^T = the transpose of V) and S contains the singular values along the diagonal.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [U,S,V] = sing_val_decomp(A)\r\n\r\n\r\nend","test_suite":"%%\r\nA = [3 0; 0 -2];\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(isequal(U,[1 0; 0 1]))\r\nassert(isequal(S,[3 0; 0 2]))\r\nassert(isequal(V,[1 0; 0 -1]))\r\n\r\n%%\r\nA = [0 2; 0 0; 0 0];\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(isequal(U,eye(3)))\r\nassert(isequal(S,fliplr(A)))\r\nassert(isequal(V,[0 -1; 1 0]))\r\n\r\n%%\r\nA = [3 1 1; -1 3 1];\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\nassert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\nassert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\r\n%%\r\nA = magic(3);\r\n[U,S,V] = sing_val_decomp(A);\r\nassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\nassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\nassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tA = [3 0; 0 -2];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,[1 0; 0 1]))\r\n\t\tassert(isequal(S,[3 0; 0 2]))\r\n\t\tassert(isequal(V,[1 0; 0 -1]))\r\n\tcase 2\r\n\t\tA = [0 2; 0 0; 0 0];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,eye(3)))\r\n\t\tassert(isequal(S,fliplr(A)))\r\n\t\tassert(isequal(V,[0 -1; 1 0]))\r\n\tcase 3\r\n\t\tA = [3 1 1; -1 3 1];\r\n                [U,S,V] = sing_val_decomp(A);\r\n                assert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\n                assert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\n                assert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\tcase 4\r\n\t\tA = magic(3);\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tA = [3 0; 0 -2];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,[1 0; 0 1]))\r\n\t\tassert(isequal(S,[3 0; 0 2]))\r\n\t\tassert(isequal(V,[1 0; 0 -1]))\r\n\tcase 2\r\n\t\tA = [0 2; 0 0; 0 0];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,eye(3)))\r\n\t\tassert(isequal(S,fliplr(A)))\r\n\t\tassert(isequal(V,[0 -1; 1 0]))\r\n\tcase 3\r\n\t\tA = [3 1 1; -1 3 1];\r\n                [U,S,V] = sing_val_decomp(A);\r\n                assert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\n                assert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\n                assert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\tcase 4\r\n\t\tA = magic(3);\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tA = [3 0; 0 -2];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,[1 0; 0 1]))\r\n\t\tassert(isequal(S,[3 0; 0 2]))\r\n\t\tassert(isequal(V,[1 0; 0 -1]))\r\n\tcase 2\r\n\t\tA = [0 2; 0 0; 0 0];\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(isequal(U,eye(3)))\r\n\t\tassert(isequal(S,fliplr(A)))\r\n\t\tassert(isequal(V,[0 -1; 1 0]))\r\n\tcase 3\r\n\t\tA = [3 1 1; -1 3 1];\r\n                [U,S,V] = sing_val_decomp(A);\r\n                assert(sum(sum(abs(U-(-[1 -1; 1 1]/sqrt(2)))))\u003c1e-5)\r\n                assert(sum(sum(abs(S-[sqrt(12) 0 0; 0 sqrt(10) 0])))\u003c1e-5)\r\n                assert(sum(sum(abs(V-([-0.40824829*[1;2;1],0.8944*[1;-0.5;0],0.1826*[-1;-2;5]]))))\u003c5e-3)\r\n\tcase 4\r\n\t\tA = magic(3);\r\n\t\t[U,S,V] = sing_val_decomp(A);\r\n\t\tassert(sum(sum(abs(U-[-0.577350*ones(3,1), 0.70710678*[1;0;-1], 0.40824829*[1;-2;1]])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(S-[15,0,0;0,6.928203,0;0,0,3.4641016])))\u003c1e-5)\r\n\t\tassert(sum(sum(abs(V-[-0.577350*ones(3,1), 0.40824829*[1;-2;1], 0.70710678*[1;0;-1]])))\u003c1e-5)\r\nend\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":4,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":37,"test_suite_updated_at":"2021-07-01T09:46:16.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-10T02:16:17.000Z","updated_at":"2026-03-19T18:40:04.000Z","published_at":"2015-03-10T02:16:17.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the three matrices of the singular value decomposition (A = U*S*V^T) for each provided matrix. U and V are square unitary matrices (V^T = the transpose of V) and S contains the singular values along the 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