PROPERTIES OF MATRICES
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1 PROPERIES OF MARICES ajoint... 4, 5 algebraic multiplicity... augmente matri... basis..., cofactor... 4 coorinate vector... 9 Cramer's rule... eterminant..., 5 iagonal matri... 6 iagonalizable... 8 imension... 6 ot prouct... 8 eigenbasis... eigenspace... eigenvalue... eigenvector... geometric multiplicity... ientity matri... 4 image... 6 inner prouct... 9 inverse matri... 5 inverse transformation.. 4 invertible... 4 isomorphism... 4 kernal... 6 Laplace epansion by minors... 8 linear inepenence... 6 linear transformation... 4 lower triangular... 6 INDEX norm... nullity... 8 orthogonal..., 9 orthogonal iagonalization... 8 orthogonal projection... orthonormal... orthonormal basis... pivot columns... quaratic form... 9 rank... reuce row echelon form... reflection... 8 row operations... rref... similarity... 8 simultaneous equations singular... 8 skew-symmetric... 6 span... 6 square... submatrices... 8 symmetric matri... 6 trace... transpose... 5, 6 BASIC OPERAIONS - aition, subtraction, multiplication For eample purposes, let A a c then A + B a b ± e c g an AB a c b e g a a scalar times a matri is c b an B e g f a ± e b ± f h c ± g ± h f ae + bg af + bh h ce + g cf + h b a b c CRAMER'S RULE for solving simultaneous equations Given the equations: We epress them in matri form: Accoring to Cramer s rule: 8 A 4 o fin we replace the first column of A with vector y an ivie the eterminant of this new matri by the eterminant of A. f h 4 A 4 an C i j o fin we replace the secon column of A with vector y an ivie the eterminant of this new matri by the eterminant of A. AC a c Where matri A is A b i + ai bj j ci + j an vector y is 8 A 4 o fin we replace the thir column of A with vector y an ivie the eterminant of this new matri by the eterminant of A. om Penick tom@tomzap.com /8/5 Page of
2 HE DEERMINAN he eterminant of a matri is a scalar value that is use in many matri operations. he matri must be square (equal number of columns an rows) to have a eterminant. he notation for absolute value is use to inicate "the eterminant of", e.g. A means "the eterminant of matri A" an a b means to c take the eterminant of the enclose matri. Methos for fining the eterminant vary epening on the size of the matri. he eterminant of a matri is simply: where A a b c, et a b A A a bc c he eterminant of a matri can be calculate by repeating the first two columns as shown in the figure at right. hen a the proucts of each of three iagonal rows an subtract the proucts of the three crossing iagonals as shown. a a a + a a a + a a a a a a a a a a a a his metho use for matrices oes not work for larger matrices. a a a a a a a a a a a a a a a he eterminant of a 4 4 matri can be calculate by fining the eterminants of a group of submatrices. Given the matri D we select any row or column. Selecting row of this matri will simplify the process because it contains a zero. he first element of row one is occupie by the number which belongs to row, column. Mentally blocking out this row an column, we take the eterminant of the remaining matri. Using the metho above, we fin the eterminant of to be 4. Proceeing to the secon element of row, we fin the value occupying row, column. Mentally blocking out row an column, we form a matri with the remaining elements. he eterminant of this matri is 6. Similarly we fin the submatrices associate with the thir an fourth elements of row. he eterminant of is -4. It won't be necessary to fin the eterminant of 4. Now we alternately a an subtract the proucts of the row elements an their cofactors (eterminants of the submatrices we foun), beginning with aing the first row element multiplie by the eterminant like this: ( ) ( ) ( ) ( 4 ) 4 8 ( 68) et D et et + et et + he proucts forme from row or column elements will be ae or subtracte epening on the position of the elements in the matri. he upper-left element will always be ae with ae/subtracte elements occupying the matri in a checkerboar pattern from there. As you can see, we in't nee to calculate 4 because it got multiplie by the zero in row, column D Aing or subtracting matri elements: om Penick tom@tomzap.com /8/5 Page of
3 AUGMENED MARIX A set of equations sharing the same variables may be written as an augmente matri as shown at right. y + z 5 + y + z + y + z 5 REDUCED ROW ECHELON FORM (rref) Reucing a matri to reuce row echelon form or rref is a means of solving the equations. In this process, three types of row operations my be performe. ) Each element of a row may be multiplie or ivie by a number, ) wo rows may echange positions, ) a multiple of one row may be ae/subtracte to another. 5 ) We begin by swapping rows an. ) hen ivie 5 row by ) hen subtract row from row II 4) An subtract times row from row (I) ) hen subtract row from row ) An ivie 5 row by II ) A.5 row to row (III) 8) An subtract row from row. 5 -(III) he matri is now in reuce row echelon form an if we rewrite the equations with these new values we have the solutions. A matri is in rref when the first nonzero element of a row is, all other elements of a column containing a leaing are zero, an rows are orere progressively with the top row having the leftmost leaing. y z When a matri is in reuce row echelon form, it is possible to tell how may solutions there are to the system of equations. he possibilities are ) no solutions - the last element in a row is non-zero an the remaining elements are zero; this effectively says that zero is equal to a non-zero value, an impossibility, ) infinite solutions - a non-zero value other than the leaing occurs in a row, an ) one solution - the only remaining option, such as in the eample above. RANK If an invertible matri A has been reuce to rref form then its eterminant can be foun by et( A ) ( ) s kk k, where s is the number of row swaps performe an k r, k, k r are the scalars by which rows have been ivie. he number of leaing 's is the rank of the matri. Rank is also efine as the imension of the largest square submatri having a nonzero eterminant. he rank is also the number of vectors require to form a basis of the span of a matri. om Penick tom@tomzap.com /8/5 Page of
4 HE IDENIY MARIX In this case, the rref of A is the ientity matri, enote I n characterize by the iagonal row of 's surroune by zeros in a square matri. When a vector is multiplie by an ientity matri of the same imension, the prouct is the vector itself, I n v v. rref( A ) LINEAR RANSFORMAION his system of equations can be represente in the form A b. his is also known as a linear transformation from to b because the matri A transforms the vector into the vector b. ADJOIN For a matri, the ajoint is: For a an higher matri, the ajoint is the transpose of the matri after all elements have been replace by their cofactors (the eterminants of the submatrices forme when the row an column of a particular element are eclue). Note the pattern of signs beginning with positive in the upper-left corner of the matri. A y z where A a b c, aj b A c a a b c where B e f, g h i 5 b e f f e e f b c b c h i g i g h h i h i e f b c a c a b f a c a c aj B h i g i g h g i g i f b c a c a b e a b a b e f f e g h g h e INVERIBLE MARICES A matri is invertible if it is a square matri with a eterminant not equal to. he reuce row echelon form of an invertible matri is the ientity matri rref(a) I n. he eterminant of an inverse matri is equal to the inverse of the eterminant of the original matri: et(a - ) /et(a). If A is an invertible n n matri then rank(a) n, im(a) R n, ker(a) {}, the vectors of A are linearly inepenent, is not an eigenvalue of A, the linear system A b has a unique solution, for all b in R n. HE INVERSE RANSFORMAION If A is an invertible matri, the inverse matri coul be use to transform b into, A b, A - b. An invertible linear transform such as this is calle an isomorphism. A A A matri multiplie by its inverse yiels the ientity matri. BB - I n om Penick tom@tomzap.com /8/5 Page 4 of
5 FINDING HE INVERSE MARIX Metho o calculate the inverse matri, consier the B invertible matri B. 8 ) Rewrite the matri, aing the ientity matri to the right. 8 ) Perform row operations on the 6 matri to put B in rref form. hree types of row operations are: ) Each element of a row may be multiplie or ivie by a number, ) wo rows may echange positions, ) a multiple of one row may be ae/subtracte to another. 6 5 ) he inverse of B is now 6 in the matri to the B right. 5 If a matri is orthogonal, its inverse can be foun simply by taking the transpose. FINDING HE INVERSE MARIX Metho o calculate the ) First we must fin the ajoint of inverse matri, matri B. he ajoint of B is the consier the invertible transpose of matri B after all matri B. elements have been replace by their cofactors. (he metho of fining B the ajoint of a matri is ifferent; see page 4.) he 8 notation means "the eterminant of". 8 8 aj B 8 8 ) Calculating the eterminants we get. aj B 6 5 ) An then taking the transpose we get. 6 aj B 5 4) Now we nee the eterminant of B. et he formula for the inverse matri is aj B B et B 5) Filling in the values, we have the solution. B om Penick tom@tomzap.com /8/5 Page 5 of
6 SYMMERIC MARICES A symmetric matri is a square matri that can be flippe across the iagonal without changing the elements, i.e. A A. All eigenvalues of a symmetric matri are real. Eigenvectors corresponing to istinct eigenvalues are mutually perpenicular A skew-symmetric matri has off-iagonal elements mirrore by their negatives across the iagonal. A -A. MISCELLANEOUS MARICES he transpose of a matri A is written A an is the n m matri whose ijth entry is the jith entry of A. A iagonal matri of equal elements commutes with any matri, i.e. AB BA. A 9 5 A A 9 5 IMAGE OF A RANSFORMAION he image of a transformation is its possible values. he image of a matri is the span of its columns. An image has imensions. For eample if the matri has three rows the image is one of the following: ) -imensional space, et(a), rank ) -imensional plane, et(a), rank ) -imensional line, et(a), rank 4) -imensional point at origin, A SPAN OF A MARIX he span of a matri is all of the linear combinations of its column vectors. Only those column vectors which are linearly inepenent are require to efine the span. KERNAL OF A RANSFORMAION he kernal of a transformation is the set of vectors that are mappe by a matri to zero. he kernal of an invertible matri is zero. he imension of a kernal is the number of vectors require to form the kernal. a b A iagonal matri is compose of zeros ecept for the iagonal an is commutative with another iagonal matri, i.e. AB BA. A lower triangular matri has s above the iagonal. Similarly an upper triangular matri has s below. Given the matri: A a c b c 5 of the transformation A, the image consists of all combinations of its (linearly inepenent) column vectors. + + ( ) A span c c c + + C 5 kernal LINEAR INDEPENDENCE A collection of vectors is linearly inepenent if none of them are a multiple of another, an none of them can be forme by summing multiples of others in the collection. om Penick tom@tomzap.com /8/5 Page 6 of
7 BASIS A basis of the span of a matri is a group of linearly inepenent vectors which span the matri. hese vectors are not unique. he number of vectors require to form a basis is equal to the rank of the matri. A basis of the span can usually be forme by incorporating those column vectors of a matri corresponing to the position of leaing s in the rref matri; these are calle pivot columns. he empty set θ is a basis of the space {}. here is also basis of the kernal, basis of the image, eigenbasis, orthonormal basis, etc. In general terms, basis infers a minimum sample neee to efine something. RACE A trace is the sum of the iagonal elements of a square matri an is written tr(a). ORHONORMAL VECORS Vectors are orthonormal if they are all unit vectors (length ) an are orthogonal (perpenicular) to one another. Orthonormal vectors are linearly inepenent. heir ot prouct of orthogonal vectors is zero. ORHOGONAL MARIX An orthogonal matri is compose only of orthonormal vectors; it has a eterminant of either or -. An orthogonal matri of eterminant is a rotation matri. Its use in a linear transformation is calle a rotation because it rotates the coorinate system. Matri A is orthogonal iff A A I n, or equivalently A - A. ORHOGONAL PROJECION V is an n m matri. v, v, v m are an orthonormal basis of V. For any vector in R n there is a unique vector w in V such that w. he ORHOGONAL PROJECION OF ONO V vector w is calle the orthogonal projection w projv ( v ) v ( v m ) v m of onto V. see also Gram-Schmit.pf EIGENVECORS AND EIGENVALUES Given a square matri A, an eigenvector is any vector v such that Av is a scalar multiple of A. he eigenvalue woul be the scalar for which this is true. Av λv. o etermine the eigenvalues, solve the characteristic polynomial et(λi n - A) for values of λ. hen convert to rref form an solve for the coefficients as though it was a matri of simultaneous equations. his forms a column vector which is an eigenvector. Where there are 's, you can let the coefficient equal. EIGENSPACE he eigenspace associate with an eigenvalue λ of an n n matri is the kernal of the matri A - λi n an is enote by E λ. E λ consists of all solutions v of the equation Av λv. In other wors, E λ consists of all eigenvectors with eigenvalue λ, together with the zero vector. EIGENBASIS An eigenbasis of an n n matri A is a basis of R n consisting of unit eigenvectors of A. o convert a vector to a unit vector, sum the squares of its elements an take the inverse square root. Multiply the vector by this value. GEOMERIC MULIPLICIY he geometric multiplicity for a given eigenvalue λ is the imension of the eigenspace E λ ; in other wors, the number of eigenvectors of E λ. he geometric multiplicity for a given λ is equal to the number of leaing zeros in the top row of rref(a - λi n ).. ALGEBRAIC MULIPLICIY he algebraic multiplicity for a given eigenvalue λ is the number of times the eigenvalue is repeate. For eample if the characteristic polynomial is (λ-) (λ-) then for λ the algebraic multiplicity is an for λ the algebraic multiplicity is. he algebraic multiplicity is greater than or equal to the geometric multiplicity. om Penick tom@tomzap.com /8/5 Page of
8 LAPLACE EXPANSION BY MINORS his is a metho for fining the eterminant of larger matrices. he process is simplifie if some of the elements are zeros. ) Select the row or column with the most zeros. ) Beginning with the first element of this selecte vector, consier a submatri of all elements that o not belong to either the row or column that this first element occupies. his is easier to visualize by rawing a horizontal an a vertical line through the selecte element, eliminating those elements which o not belong to the submatri. ) Multiply the eterminant of the submatri by the value of the element. 4) Repeat the process for each element in the selecte vector. 5) Sum the results accoring to the rule of signs, that is reverse the sign of values represente by elements whose subscripts i & j sum to an o number. DIAGONALIZABLE If an n n matri has n istinct eigenvalues, then it is iagonalizable. NULLIY he nullity of a matri is the number of columns in the result of the matlab comman null(a). SINGULAR MARIX A singular matri is not invertible. SIMILARIY Matri A is similar to matri B if S - AS B. Similar matrices have the same eigenvalues with the same geometric an algebraic multiplicities. heir eterminants, traces, an rank are all equal REFLECION Given that L is a line in R n, v is a vector in R n an u is a unit vector along L in R n, the reflection of v in L is: DO PRODUC he ot prouct of two matrices is equal to the transpose of the first matri multiplie by the secon matri. A B A B Eample: [ ] ORHOGONAL DIAGONALIZAION A matri A is iagonalizable if an only if A is symmetric. D S AS where D is a iagonal matri whose iagonal is compose of the eigenvalues of A with the remainer of the elements equal to zero, S is an orthogonal matri whose column vectors form the eigenbasis of A. o fin D we nee only fin the eigenvalues of A. o fin S we fin the eigenvectors of A. If A has istinct eigenvalues, the unit eigenvectors form S, otherwise we have more work to o. For eample if we have a matri with eigenvalues 9,,, we first fin a linearly inepenent eigenvector for each eigenvalue. he eigenvector for λ 9 (we'll call it y) will be unique an will become a vector in matri S. We must choose eigenvectors for λ so that one of them is orthogonal (we'll call it ) to the eigenvector y from λ 9, by keeping in min that the ot prouct of two orthogonal vectors is zero. he remaining non-orthogonal eigenvector from λ 9 we will call v. Now from the eigenspace, v we must fin an orthogonal vector to replace v. Using the formula for orthogonal projection w projv ( v ) v, we plug in our values for an v an obtain vector w, orthogonal to. Now matri S [w y]. We can check our work by performing the calculation S - AS to see if we get matri D. PRINCIPLE SUBMARICES Give a matri: , the principle submatrices are: [ ], ref v (proj v) v ( u v) u v L 4 L, an om Penick tom@tomzap.com /8/5 Page 8 of
9 om Penick /8/5 Page 9 of
10 COORDINAE VECOR If we have a basis B consisting of vectors b, b, b n, then any vector in R n can be written as: c he vector c c is the coorinate vector of an: M c n c b + c b + L+ Bc Determining the Coorinate Vector Given B an, we fin c by forming an augmente matri from B an, taking it to rref form an reaing c from the right-han column. QUADRAIC FORM A function such as q ( ) q(, ) is calle a quaratic form an may be written in the form q( ) A. Notice in the eample at right how the - term is split in half an use to form the "symmetric" part of the symmetric matri. POSIIVE DEFINIE: Matri A is positive efinite if all eigenvalues are greater than, in which case q() is positive for all nonzero, an the eterminants of all principle submatrices will be greater than. NEGAIVE DEFINIE: Matri A is negative efinite if all eigenvalues are less than, in which case q() is negative for all nonzero. INDEFINIE: Matri A is inefinite if there are negative an positive eigenvalues in which case q() may also have negative an positive values. What about eigenvalues which inclue? he efinition here varies among authors. DISANCE OF WO ELEMENS OF AN INNER PRODUC ist ( f, g c n Eample: q ( ) q(, ) q( ) A b 6 q( ) A 8 ) f g b n [ f ( t) g( t) ] a t INNER PRODUC An inner prouct in a linear space V is a rule that assigns a real scalar (enote by f, g to any pair f, g of elements of V, such that the following properties hol for all f, g, h in V, an all c in R. A linear space enowe with an inner prouct is calle an inner prouct space. a. f, g g, f b. f + g, h f, h + g, h c. cf, g c f, g. f, f > for all nonzero f in V. wo elements f, g of an inner prouct space are orthogonal if: f, g om Penick tom@tomzap.com /8/5 Page of
11 NORM he norm of a vector is its length: v v + v + L + v n he norm of an element f of an inner prouct space is: f f, f a b f t om Penick tom@tomzap.com /8/5 Page of
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