Matrices - PROPERTIES OF MATRICES INDEX adjoint.4, 5...

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Tom Penick tomzap@eden.com www.teicontrols.com/notes 2/29/2000 Page 1 of 10 PROPERTIES OF MATRICES INDEX adjoint. ...................... 4, 5 algebraic multiplicity . ....7 augmented matrix. ........ 3 basis. ........................ 3, 7 cofactor . ....................... 4 coordinate vector . ......... 9 Cramer's rule. ............... 1 determinant. .............. 2, 5 diagonal matrix . ............ 6 diagonalizable. .............. 8 dimension. .................... 6 dot product . .................. 8 eigenbasis . ................... 7 eigenspace. .................. 7 eigenvalue . ................... 7 eigenvector. .................. 7 geometric multiplicity. ...7 identity matrix. .............. 4 image . .......................... 6 inner product. ............... 9 inverse matrix. .............. 5 inverse transformation. .4 invertible. ...................... 4 isomorphism. ................ 4 kernal . .......................... 6 Laplace expansion by minors . .................... 8 linear independence. ....6 linear transformation. ....4 lower triangular. ............ 6 norm . ......................... 10 nullity. ........................... 8 orthogonal. ............... 7, 9 orthogonal diagonalization ................................ 8 orthogonal projection. ... 7 orthonormal. ................. 7 orthonormal basis . ....... 7 pivot columns. .............. 7 quadratic form. ............. 9 rank. ............................. 3 reduced row echelon form ................................ 3 reflection . ..................... 8 row operations . ............ 3 rref. ............................... 3 similarity . ...................... 8 simultaneous equations1 singular. ........................ 8 skew-symmetric. ........... 6 span . ............................ 6 square . ......................... 2 submatrices. ................. 8 symmetric matrix . ......... 6 trace. ............................ 7 transpose. ................. 5, 6 BASIC OPERATIONS - addition, subtraction, multiplication For example purposes, let A = a b c d and B = e f g h and C = i j then A B + = ± = ± ± ± ± a b c d e f g h a e b f c g d h and AB = = + + + + a b c d e f g h ae bg af bh ce dg cf dh AC = = + + a b c d i j ai bj ci dj a scalar times a matrix is 3 3 3 3 3 a b c d a b c d = CRAMER'S RULE for solving simultaneous equations Given the equations: 3 2 3 2 1 = + + x x x 7 3 3 2 1 = - + x x x 1 3 2 1 = + + x x x We express them in matrix form: = - 1 7 3 1 1 1 1 3 1 1 1 2 3 2 1 x x x Where matrix A is - = 1 1 1 1 3 1 1 1 2 A and vector y is 1 7 3 According to Cramer’s rule: 1 311 7 31 111 8 2 4 x A - = == To find x 1 we replace the first column of A with vector y and divide the determinant of this new matrix by the determinant of A . 2 231 1 71 4 1 4 x A - = To find x 2 we replace the second column of A with vector y and divide the determinant of this new matrix by the determinant of A . 3 213 137 8 2 4 x A - = = =- To find x 3 we replace the third column of A with vector y and divide the determinant of this new matrix by the determinant of A .
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Tom Penick tomzap@eden.com www.teicontrols.com/notes 2/29/2000 Page 2 of 10 THE DETERMINANT The determinant of a matrix is a scalar value that is used in many matrix operations. The matrix must be square (equal number of columns and rows) to have a determinant. The notation for absolute value is used to indicate "the determinant of", e.g.
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Matrices - PROPERTIES OF MATRICES INDEX adjoint.4, 5...

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