properties of determinant?
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There are many general properties of determinants of matrices for example\n\nConsider two matrices A and B then\ndet(AB)= det(A) det(B)\n\ndet(A) = det( Transpose(A) )\n\nIf you interchange two rows in a matrix then you will change the sign of the determinant.  Interchanging two columns does the same thing.\n\nThere are many more but these should get you started
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I assume you mean invertible, not convertible.\n\nIt depends on how detailed you want to get. If you do not want to dig in extremely deeply, you can just do it by construction. \n\n1. A symmetric matrix M has an orthogonal decomposition\n\nM=O'DO\n\nwhere O is an orthogonal matrix, O'O=I, the identity, and D is a diagonal matrix with eigenvalues of M down the diagonal.\n\n2. If M is positive definite, then the diagonal elements of D are >0\n\n3. Produce the matrix \n\nA=O'(D^(-1))O\n\n\n4. Verify that A is the inverse of M\n\ni.e., \n\nA*M=M*A=I
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Try this:\nhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut9_bar.htm
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If A is an n-by-n symmetric positive definite matrix with real entries, that means ((v transpose)Av) > 0 for nonzero real n-by-1 column vectors v.\n\nConsider the 3x3 case with A equal to\n\n[ a b c ]\n[ b d e ]\n[ c e f ]\n\nLet u, v, and w be the nonzero column vectors\n\n[ 1 ]\n[ 0 ]\n[ 0 ]\n\n[ 0 ]\n[ 1 ]\n[ 0 ]\n\n[ 0 ]\n[ 0 ]\n[ 1 ]\n\nrespectively.  Then\n\n((u transpose)Au) = a > 0\n((v transpose)Av) = d > 0\n((w transpose)Aw) = f > 0\n\nCan you see how to generalize this for other values of n besides three and/or for matrices and vectors with complex entries?
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ax^2 + bx + c = 0\n\nx = [-b +/- sqrt(b^2 - 4ac)]/ 2a\ntwo answers since there is a +/- in there, and if the determinant (b^2 - 4ac) is negative, then the answers are complex conjugate pairs.\n\n1) x = 2 +/- sqrt(4-3) = 1,3 [real distinct]\n2) x = (-3 +/- sqrt(9 - 40))/4\n= -0.75 +/- i * sqrt(31/16)\n= -0.75 +/- i * 1.39 [complex conjugate pair]
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1. Write down a couple of Jordan blocks.\n\n2. Write down their squares, cubes, maybe even a fourth power.\n\n3. Gaze at each set of identity matrix, J, J^2, J^3, ... until you see a simple linear combination of the matrices that sums to the all zero matrix.  That is your minimal polynomial.\n\n4. Generalize.\n\n5. Now it is obvious. :-)\n\nAlternative:\n\n1.  Google for some combination of:\n\nminimal polynomial Jordan block\n"minimal polynomial" "Jordan block"\nJordan block minimal polynomial\n"Jordan block" "minimal polynomial"\n\n2. Read answer.\n\n3. Too bad, it still isn't obvious. :-(
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The principal diagonal is the most useful diagonal in the study of matrix algebra because it is related to several invariants. \nIf M and N are two matrices of the same linear transformation with respect to different bases, then their trace (the sum of the entries on the principal diagonal) is the same.\nA transformation that can be represented as a scaling of n independent vectors in an n-dimensional space has a diagonal representation in that space under a suitable change of bases (namely to those vectors it scales, called eigenvectors). The product of the entries of the diagonal in that representation is another invariant, called the determinant (it is the signed volume of a parallelepiped made up of the basis vectors after the transformation). The determinant is another important invariant that you will see all over the place in differential geometry.\nThe other diagonals do have names (the superdiagonal is on top of the principal, subdiagonal is on the bottom of the principal, etc.) but they are so rarely important that they have fallen out of fashion. See the source for more details.
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Example:\nIf z=f(u,v),\nu=g(x,y) and\nv=h(x,y), then\n(all derivatives are supposed to be partial derivatives)\ndz/dx=dz/du*du/dx+dz/dv*dv/dx\ndz/dy=dz/du*du/dy+dz/dv*dv/dy\n\nAlternatively, you can set of the matrices\n[[dz/du dz/dv]]\nand\n[[ du/dx du/dy]\n[dv/dx dv/dy]]\nand get that\n[[dz/dx dz/dy]]\nis the product of the two matrices.
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Check out this web page: http://mathworld.wolfram.com/Net.html
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CIRCLE CHARACTERISTICS:\na) the equation contains the variable x & y.\nb) the highest degree for x is 2; the highest degree...\n...for y is 2.\nc) the coefficients of x^2 and y^2 are the same.\nd) the operation between x^2 and y^2 is plus (+),...\n...provided the sign before the x^2 is positive (+).\n\nSTANDARD EQUATION OF A CIRCLE:\n\n(x-h)^2 + (y-k)^2 = r^2 \nwhere:\nh = x-coordinate of the center of the circle.\nk = y-coordinate of the center of the circle.\nr = radius of the circle.\n\nEXAMPLE: \nProve that 4x^2 + 4y^2 - 24x + 32y - 44 = 0 is an equation of a circle.\n\nPROOF:\n\n4x^2 + 4y^2 - 24x + 32y - 44 = 0 ....................Given\n4x^2 - 24x + 4y^2 + 32y - 44 = 0 ...............Commutative\nx^2 - 6x + y^2 + 8y - 11 = 0....................Divide by 4\n(x^2 - 6x) + (y^2 + 8y) - 11 = 0................Associative\n{x^2 - 6x + 9) + (y^2 +8y +16) - 11 - 9 - 16 = 0  ......... \n..............Completing the Square\n(x^2 - 6x + 9) + (y^2 + 8y + 16) - 36 = 0 .....Subtraction\n(x^2 - 6x + 9) + (y^2 + 8y + 16) = 36 .......Transposition\n(x - 3)^2 + (y + 4)^2 = 36 ........Trinomial Perfect Square\n(x - 3)^2 + [y -(-4)]^2 = 6^2 ...Std. Equation of Circle\nwhere:\nh = 3; k = -4; r = 6\nMeaning: The above equation is an equation of a circle with center at (3,-4) and radius, r = 6.\n\nTherefore, since...\n4x^2 + 4y^2 - 24x + 32y - 44 = 0 can be transformed into a standard equation of a circle & having the characteristics of a circle, then:\n\nConclusion:\nThe equation 4x^2 + 4y^2 - 24x + 32y - 44 = 0 must be that of a circle.
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Homework.\n\nLook at B=A. You should find something interesting about tr(A^tA).
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|10 . -9 . -8|\n|. 5 . -1 . -3|\n|. 0 . -4 . -6|\n\ncofactors are\n\n| +. - + |\n|. - +. - |\n| +. - + |\n\nManipulate the determinant\n\nRow 1 + 5 x row 2\nRow 3 - 3 x row 2\nyields\n\n|-35. -9 19|\n| . 0 .  -1 . 0| =\n|-20 .-4 ..6|\n\n= + -1(-35 x 6 - -20 x 19) on expanding along middle row\n\n= 210 - 380\n\n= -170
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distributive property;\n\na(b+c) = ab + ac
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Basically, Cramer's Rule lets you solve for a variable in a system of equations via matrix operations without solving the whole matrix.  The process requires some knowledge of finding determinants.  The site below walks through an example of how this works.
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In the following: x, y, z real vectors and c real scalar\n\n\n- scalar product: x.y = x1*y1 + x2*y2 + x3*y3\n\n- euclidian distance: √x.y\n\n\nThere are 4 properties deriving from scalar product:\n\n\nsymmetry\n\n1: x.y == y.x\n\n\nlinearity\n\n2: (x+z).y == x.y + z.y\n\n3: (c*x).y == c(x.y) == x.(c*y)\n\n\npositive definiteness\n\n4: for all x: x.x ≥ 0 and  x.x == 0 iff x == 0\n\n\n\nEuclidian distance is a specific definition of a vector norm || . ||:\n\nEuclidian norm: ||x|| =  √x.x\n\n\nThe following holds for each scalar product induced norm: \n\nSchwarz's inequality\n\n5: |x.y| ≤ ||x||*||y||\n\n\nTriangle inequality\n\n6: | ( ||x|| - ||y|| ) | ≤ ||x + y|| ≤ ||x|| + ||y||\n\n\n\nRemark: | . | acts on real numbers and || . || on vectors
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The formula for the inverse of a matrix also requires the determinant of the whole matrix. Minors are determinants of submatrices where one row and one column are deleted. See the source for the equation.
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associative, commutative, identity, inverse