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──────────────────────────────────────────────────────────── ──────────────────────────────────────────────────────────── DOCTOR MATRIX Version 1.00 ──────────────────────────────────────────────────────────── Shareware Vector and Matrix Algebra Programming Package for C++ ──────────────────────────────────────────────────────────── FUNCTION REFERENCE ──────────────────────────────────────────────────────────── by Zvika Ben-Haim ──────────────────────────────────────────────────────────── ──────────────────────────────────────────────────────────── ──────────────────────────────────────────────────────────── TABLE OF CONTENTS ──────────────────────────────────────────────────────────── GENERAL NOTES VECTOR OPERATORS Addition Assignment (=) Equality Input (>>) Multiplication by Scalar Multiplication (Dot Product) Output (<<) Subscript Subtraction (Binary) Subtraction (Unary) VECTOR MEMBER FUNCTIONS Absolute Value Alignment Constructors Destructor Leading Element Length (Order) Transpose MATRIX OPERATORS Addition Assignment (=) Equality Input (>>) Multiplication by Scalar Multiplication (Matrix by Matrix) Output (<<) Subscript Subtraction (Binary) Subtraction (Unary) MATRIX MEMBER FUNCTIONS Canonized Matrix, Equivalent Column Constructors Destructor Determinant Equivalent Matrices, see Canonized and Stepped Matrices Invert Leading Element Number of Columns Number of Rows Rank Row Stepped Matrix, Equivalent Trace Transpose MATRIX PROPERTIES Antisymmetric Matrix Diagonal Matrix Identity Matrix Regular (Invertible) Matrix Scalar Matrix Singular (Non-Invertible) Matrix Square Matrix Stepped Matrix Symmetric Matrix LIST OF ERROR MESSAGES ──────────────────────────────────────────────────────────── GENERAL NOTES ──────────────────────────────────────────────────────────── The following list describes all of the functions and operators supported by Doctor Matrix. Basic familiarity with Doctor Matrix classes is assumed. It is recommended that you read the Programmer's Guide before using the more advanced functions described here. Furthermore, a basic understanding of the C++ class mechanism is highly recommended. Throughout this document, the word `scalar', where it refers to a data type, is to be understood as an object of the data type of the elements of the matrix or vector in question. In general, whenever an error is encountered, a message is sent to the `cerr' stream. When such an error is reported, the value returned by the function is meaningless, and is to be ignored. However, all such cases can effectively be avoided by checking the validity of the parameters passed to the function before calling it. For instance, to avoid the `Subscript out of range' error, the programmer should call the member functions len() (for vectors) or nrows() and ncols() (for matrices), and ensure that the subscripts are within the range returned by these functions. The only exception is the "out of memory" errors, which cause the program to abort. A full list of errors and their explanations is supplied at the end of this document. ──────────────────────────────────────────────────────────── VECTOR OPERATORS ──────────────────────────────────────────────────────────── ADDITION ──────────────────────────────────────────────────────────── Syntax: v1+v2 Operands:v1 and v2 must be vectors of the same order and data type Returns: The sum of v1 and v2 Notes: The sum of two vectors is defined as the sum of their respective elements. In particular, if S=v1+v2, then S(i)=v1(i)+v2(i) for all i. ASSIGNMENT ──────────────────────────────────────────────────────────── Syntax: v1=v2; Operands:v1 and v2 must be vectors of the same type Returns: v2 is copied into v1, and is also the return value Notes: v1's new alignment, order and elements are equal to those of v2. The effect of this operator is equal in all respects to the effects of Form 4 of the constructor. The operator is required for reasons of internal memory management. EQUALITY ──────────────────────────────────────────────────────────── Syntax: v1==v2 Operands:v1 and v2 must be vectors of the same data type Returns: TRUE (1) if v1 is equal to v2; FALSE (0) otherwise Notes: Two vectors v1 and v2 are said to be equal if they are of the same order and alignment and if all elements in v1 are equal to their corresponding elements in v2. In other words, v1(i)==v2(i) for all i. INPUT (>>) ──────────────────────────────────────────────────────────── Syntax: istr >> v; Operands:istr is any object of type istream or derived types, e.g. cin v is any vector Returns: istr Notes: The contents of v are overwritten with values entered from the specified stream. The number of items read from the stream is equal to the order of the vector. The vector preserves its original alignment. MULTIPLICATION BY SCALAR ──────────────────────────────────────────────────────────── Syntax: a*v Operands:v is any vector a is any number of data type equal to that of the vector's elements, or a type that can be implicitly cast to the vector's elements (e.g. a Vector<float> can be multiplied by ints) Returns: The product of a and v Notes: The product of a scalar by a vector is a vector of the same order and alignment as the original, in which all elements are equal to the original elements multiplied by the scalar. For instance, if P=n*V then P(i)=n*V(i) for all i. MULTIPLICATION (DOT PRODUCT) ──────────────────────────────────────────────────────────── Syntax: v1*v2 Operands:v1 and v2 are vectors of the same data type and order v1 is aligned horizontally and v2 is aligned vertically Returns: The dot product v1.v2, which is a scalar of the same type as the elements of v1 and v2 Notes: The scalar product (or dot product) of v1 and v2 is equal to |v1|*|v2|*cos , where |vn| is the length of the vector vn, and is the angle between v1 and v2. It can be shown that this is also equal to: v1(1)*v2(1) + v1(2)*v2(2) + ... + v1(n)*v2(n) where n is the order of the two vectors. Notice that the vector alignment must be as defined above, otherwise an error is generated. OUTPUT (<<) ──────────────────────────────────────────────────────────── Syntax: ostr << v; Operands:ostr is any object of type ostream or derived types, e.g. cout v is any vector Returns: ostr Notes: The elements of v are displayed, starting at the current cursor position. The elements are separated by `\n' (for vertically aligned vectors) or `\t' (for horizontally aligned vectors). No delimiter is added after the final element. SUBSCRIPT ──────────────────────────────────────────────────────────── Syntax: v(i) Operands:v is any vector i is an integer between 1 and the order of v Returns: Reference to the i-th element in v Notes: The first element is v(1), as opposed to C/C++ arrays, in which the first element is a[0]. SUBTRACTION (BINARY) ──────────────────────────────────────────────────────────── Syntax: v1-v2 Operands:v1 and v2 must be vectors of the same order and data type Returns: The difference between v1 and v2 Notes: The difference between two vectors is defined as the difference between their respective elements. In particular, if S=v1-v2, then S(i)=m1(i)-m2(i) for all i. SUBTRACTION (UNARY) ──────────────────────────────────────────────────────────── Syntax: -v Operands:v can be any vector type Returns: The additive inverse of v, which is a vector of the same type, order and alignment Notes: The additive inverse is a vector vtag, such that v+vtag is equal to a vector containing only zeroes. It follows that for all i, v(i)=-vtag(i). ──────────────────────────────────────────────────────────── VECTOR MEMBER FUNCTIONS ──────────────────────────────────────────────────────────── ABSOLUTE VALUE ──────────────────────────────────────────────────────────── Syntax: v.abs() Parameters: None Returns: Scalar equal to the length (or absolute value) of v Notes: The abosulte value of a vector is equal to the following: sqrt(v(1)*v(1) + v(2)*v(2) + ... + v(n)*v(n)) The geometric meaning of this value is the length of the vector v. This member function can be used only if MATH_FUNCS is defined. See the Programmer's Guide for more information on the MATH_FUNCS setting. ALIGNMENT ──────────────────────────────────────────────────────────── Syntax: v.align() Parameters: None Returns: If v is aligned horizontally (default), return value is V_HORIZ (0). Otherwise, return value is V_VERT (1). Notes: The return value is of an enumeration type called V_ALIGN. CONSTRUCTORS ──────────────────────────────────────────────────────────── Syntax: Form 1: Vector<type> v(order); Form 2: Vector<type> v(order, V_VERT); Form 3: Vector<type> v(order, V_HORIZ); Form 4: Vector<type> v=v2; Parameters: type is any numeric data type (such as float, int, etc.) or any numeric class (see below) order is an integer specifying the number of elements in the vector V_VERT (1) or V_HORIZ (0) can optionally be added to specify the vector's alignment. If no alignment is specified, the vector is assumed to be aligned horizontally (V_HORIZ). The enumeration V_ALIGN defines the two alignment types. In Form 4, v2 must be of the same type as the new vector. The new vector will have the order, alignment and exact same elements as the original. Notes: Numeric classes used as data types must support arithmetic operators, assignment operators (=, +=, etc.), the equality operator (==), and stream operators (<< and >>). They must also support implicit casting from type int. If MATH_FUNCS is included, an sqrt() function receiving and returning an object of this type must also be supplied. DESTRUCTOR ──────────────────────────────────────────────────────────── The destructor is invoked transparently by the compiler when a static Vector object goes out of scope or when a dynamic Vector object is deleted. The destructor frees dynamic memory which is internally used by the Vector class. LEADING ELEMENT ──────────────────────────────────────────────────────────── Syntax: lead() Parameters: None Returns: Integer specifying the location of the leading element in v Notes: The leading element in a vector is the first nonzero element, going from left to right (or from top to bottom, in vertically aligned vectors). LENGTH (ORDER) ──────────────────────────────────────────────────────────── Syntax: v.len() Parameters: None Returns: Integer specifying the order (or number of elements) of the vector v Notes: The last element in v is accessed using v(v.len()). TRANSPOSE ──────────────────────────────────────────────────────────── Syntax: v.T() Parameters: None Returns: A vector which is the transpose of v Notes: The transposed vector is of the same type and order as v, and contains the same elements in the same order, but its alignment is opposite: if v is aligned vertically, then v.T() is aligned horizontally, and vice versa. ──────────────────────────────────────────────────────────── MATRIX OPERATORS ──────────────────────────────────────────────────────────── ADDITION ──────────────────────────────────────────────────────────── Syntax: m1+m2 Operands:m1 and m2 must be matrices of the same order and data type Returns: The sum of m1 and m2 Notes: The sum of two matrices is defined as the sum of their respective elements. In particular, if S=m1+m2, then S(i,j)=m1(i,j)+m2(i,j) for all i,j. ASSIGNMENT ──────────────────────────────────────────────────────────── Syntax: m1=m2; Operands:m1 and m2 must be matrices of the same type Returns: m2 is copied into m1, and is also the return value Notes: m1's new alignment, order and elements are equal to those of m2. The effect of this operator is equal in all respects to the effects of Form 2 of the constructor. The operator is required for reasons of internal memory management. EQUALITY ──────────────────────────────────────────────────────────── Syntax: m1==m2 Operands:m1 and m2 must be matrices of the same data type Returns: TRUE (1) if m1 is equal to m2; FALSE (0) otherwise Notes: Two matrices m1 and m2 are said to be equal if they are of the same order and if all elements in m1 are equal to their corresponding elements in m2. In other words, m1(i,j)==m2(i,j) for all i,j. INPUT (>>) ──────────────────────────────────────────────────────────── Syntax: istr >> m; Operands:istr is any object of type istream or derived types, e.g. cin m is any matrix Returns: istr Notes: The elements of m are overwritten with values entered from the specified stream. The matrix preserves its original order. If the matrix has i rows and j columns, the total number of items read from istr is i*j. MULTIPLICATION BY SCALAR ──────────────────────────────────────────────────────────── Syntax: a*m Operands:m is any matrix a is any number of data type equal to that of the matrix elements, or a type that can be implicitly cast to the matrix elements type (e.g. a Vector<float> can be multiplied by ints) Returns: The product of a and m Notes: The product of a scalar by a matrix is a matrix of the same order as the original, in which all elements are equal to the original respective elements multiplied by the scalar. For instance, if P=n*M then P(i,j)=n*M(i,j) for all i,j. MULTIPLICATION (MATRIX BY MATRIX) ──────────────────────────────────────────────────────────── Syntax: m1*m2 Operands:m1 and m2 are matrices of the same type, and the number of columns in m1 is equal to the number of rows in m2 Returns: A matrix containing the product of the two matrices. The type of the matrix is the same as the type of the original matrices. The number of rows in the result is equal to the number of rows in m1, and the number of columns in the result is equal to the number of columns in m2. Notes: If R=m1*m2, then R(i,j) is given by: R(i,j) = m1(i,1)*m2(1,j) + m1(i,2)*m2(2,j) + ... + m1(i,k)*m2(k,j) where k is the number of columns in m1, and also the number of rows in m2. OUTPUT (<<) ──────────────────────────────────────────────────────────── Syntax: ostr << m; Operands:ostr is any object of type ostream or derived types, e.g. cout m is any matrix Returns: ostr Notes: The elements of m are displayed, starting at the current cursor position. The elements in each row are separated by `\t', and the rows are separated from each other by `\n'. No delimiter is added after the final element. SUBSCRIPT ──────────────────────────────────────────────────────────── Syntax: m(i,j) Operands:m is any matrix i is an integer between 1 and the number of rows in m j is an integer between 1 and the number of columns in m Returns: Reference to element in the i-th row and j-th column in m Notes: The first element is m(1,1), and not m(0,0). SUBTRACTION (BINARY) ──────────────────────────────────────────────────────────── Syntax: m1-m2 Operands:m1 and m2 must be matrices of the same order and data type Returns: The difference between m1 and m2 Notes: The difference between two matrices is defined as the difference between their respective elements. In particular, if S=m1-m2, then S(i,j)=m1(i,j)- m2(i,j) for all i,j. SUBTRACTION (UNARY) ──────────────────────────────────────────────────────────── Syntax: -m Operands:m can be any matrix Returns: The additive inverse of m, which is a matrix of the same type and order Notes: The additive inverse is a matrix mtag, such that m+mtag is equal to a mastrix containing only zeroes. It follows that for all i and j, m(i,j)=- mtag(i,j). ──────────────────────────────────────────────────────────── MATRIX MEMBER FUNCTIONS ──────────────────────────────────────────────────────────── CANONIZED MATRIX, EQUIVALENT ──────────────────────────────────────────────────────────── Syntax: m.canon() Parameters: None Returns: The equivalent, canonical matrix Notes: A canonical matrix is a matrix which satisfies the follows: (a) it is stepped (see the step() function); (b) all leading elements are 1; (c) a column containing a leading element can contain only zeroes, except for the leading element itself. Two matrices are said to be equivalent if one can be obtained from the other using a series of allowed operations. Allowed operations are operations of the following forms: (1) Rn = Rn + k*Ri (2) Rn and Ri are switched Where Rn and Ri are the n-th and i-th rows in the matrix (treated as vectors), and k is any real number. Any matrix has only one equivalent canonical matrix, so that two matrices are equivalent if, and only if, their canonical matrices are identical. COLUMN ──────────────────────────────────────────────────────────── Syntax: m.col(i) Parameters: i is an integer specifying the column number to be returned Returns: Vector containing the elements of column i Notes: The returned vector is a copy of the elements in the i-th column. Changing the returned vector does not alter the actual elements in the original matrix. CONSTRUCTORS ──────────────────────────────────────────────────────────── Syntax: Form 1: Matrix<type> m(i,j); Form 2: Matrix<type> m = mtag; Parameters: type is any allowed data type i is an integer specifying the number of rows j is an integer specifying the number of columns mtag is any matrix of the same type as the new matrix Notes: In Form 1, a new matrix is allocated and its elements are uninitialized. In Form 2, a new matrix is created having the same order as the specified matrix, and containing elements equal to their respective elements in mtag. DESTRUCTOR ──────────────────────────────────────────────────────────── The destructor is invoked transparently by the compiler when a static Vector object goes out of scope or when a dynamic Vector object is deleted. The destructor frees dynamic memory which is internally used by the Vector class. DETERMINANT ──────────────────────────────────────────────────────────── Syntax: m.det() Parameters: None Returns: The determinant of m (of type equal to the type of the matrix elements) Notes: The determinant of a square matrix of order 2-by-2 is equal to: m(1,1)*m(2,2) - m(1,2)*m(2,1) The determinant of a square matrix of order n-by-n is defined recursively as being equal to: m(1,1)*mwithout11.det() - m(2,2)*mwithout22.det() + + ... + m(n,n)*mwithoutnn.det() where mwithoutkk is a matrix equal to m, but with the k-th row and k-th column removed. Notice that the signs of the sum are alternating. INVERT ──────────────────────────────────────────────────────────── Syntax: m.inv() Parameters: m is any square, regular matrix Returns: The inverse of matrix m Notes: If q is the inverse of m, then q*m=m*q=I, where I is the identity matrix of that order. LEADING ELEMENT ──────────────────────────────────────────────────────────── Syntax: m.lead(i) Parameters: m is any matrix with x rows i is an integer between 1 and x Returns: The location of the leading element in row i Notes: The leading element of a row is the first nonzero element in that row. NUMBER OF COLUMNS ──────────────────────────────────────────────────────────── Syntax: m.ncols() Parameters: None Returns: Number of columns in m NUMBER OF ROWS ──────────────────────────────────────────────────────────── Syntax: m.nrows() Parameters: None Returns: Number of rows in m RANK ──────────────────────────────────────────────────────────── Syntax: m.rank() Parameters: None Returns: The rank of the matrix (integer) Notes: The rank of a matrix is the number of independent rows in that matrix. A set of rows R(1), R(2),... R(n) is said to be independent if there are no coefficients a(1), a(2),... a(n), such that one of the rows (for instance R(n)) can be described as: R(n) = a(1)*R(1) + a(2)*R(2) + ... a(n-1)*R(n-1) where each row is considered to be a vector. It can be shown that all equivalent matrices have the same rank (but not all matrices of the same rank are equivalent). If you have already found an equivalent steppeor canonical matrix of the matrix m, checking the rank on the equivalent matrix (rather than on m itself) will give the same result and work much faster. ROW ──────────────────────────────────────────────────────────── Syntax: m.row(i) Parameters: i is an integer specifying the row number to be returned Returns: Vector containing the elements of row i Notes: The returned vector is a copy of the elements in the i-th row. Changing the returned vector does not alter the actual elements in the original matrix. STEPPED MATRIX, EQUIVALENT ──────────────────────────────────────────────────────────── Syntax: m.step() Parameters: None Returns: An equivalent, stepped matrix Notes: A stepped matrix is a matrix in which: (a) all rows containing only zeroes are at the bottom of the matrix (in the highest-index rows); (b) for every row except zero rows, the leading element is at least one place to the right of the leading element in the previous row. Two matrices are said to be equivalent if one can be obtained from the other using a series of allowed operations. Allowed operations are operations of the following forms: (1) Rn = Rn + k*Ri (2) Rn and Ri are switched Where Rn and Ri are the n-th and i-th rows in the matrix (treated as vectors), and k is any real number. Finding an equivalent, stepped matrix is useful, among other things, to determine matrix rank, to solve linear equations and to invert matrices. Equivalent matrices have the same rank, and stepped matrices are ranked very quickly compared to non-stepped matrices. If you intend to find both the rank of a matrix and an equivalent stepped matrix, perform the rank check on the stepped matrix to increase execution speed. TRACE ──────────────────────────────────────────────────────────── Syntax: m.tr() Parameters: None Returns: The trace of the given matrix Notes: The trace of a square matrix is defined as the sum of the diagonal elements (those having a row index equal to their column index). TRANSPOSE ──────────────────────────────────────────────────────────── Syntax: m.T() Parameters: None Returns: The transposed matrix of m Notes: The transpose of a matrix is a matrix in which each row was rewritten as a column, and vice versa. Thus element (i,j) in m will be found as element (j,i) in m.T(). Writing values into m.T() has no effect. To change the values in the transposed matrix, use the syntax: Matrix m2 = m.T(); ──────────────────────────────────────────────────────────── MATRIX PROPERTIES ──────────────────────────────────────────────────────────── All of these functions take no parameters, and return a value of type BOOL. A return value of TRUE indicates the matrix possesses the specified property. A return value of FALSE specifies the matrix does not possess the specified property, including cases where matrices of that type cannot possess the specified property (e.g. diag() for a non-square matrix). ANTISYMMETRIC MATRIX ──────────────────────────────────────────────────────────── Syntax: m.antisym() Returns TRUE if the specified matrix is antisymmetric. An antisymmetric matrix is a square matrix that satisfies the following property: For all i and j, m(i,j)=m(j,i). It follows from this that the diagonal of such a matrix contains only zeroes. DIAGONAL MATRIX ──────────────────────────────────────────────────────────── Syntax: m.diag() Returns TRUE if the specified matrix is diagonal. A diagonal matrix is a square matrix that contains zeroes in all elements except for the diagonal, that is, except for the elements of the form m(i,i) for all i. IDENTITY MATRIX ──────────────────────────────────────────────────────────── Syntax: m.iden() Returns TRUE if the specified matrix is the identity matrix. The identity matrix is a diagonal matrix containing only the number 1 in its diagonal. REGULAR MATRIX ──────────────────────────────────────────────────────────── Syntax: m.reg() Returns TRUE if the specified matrix is square and regular. A regular matrix is one that has an inverse. SCALAR MATRIX ──────────────────────────────────────────────────────────── Syntax: m.scalar() Returns TRUE if the specified matrix is scalar. A scalar matrix is a matrix that can be obtained by multiplying the identity matrix by a scalar. SINGULAR MATRIX ──────────────────────────────────────────────────────────── Syntax: m.sing() Returns TRUE if the matrix is either non-square or singular. A singular matrix is one that has no inverse. SQUARE MATRIX ──────────────────────────────────────────────────────────── Syntax: m.square() Returns TRUE if the specified matrix is square. A square matrix has the same number of rows and columns. STEPPED MATRIX ──────────────────────────────────────────────────────────── Syntax: m.stepped() Returns TRUE if the specified matrix is stepped. A matrix is stepped if it satisfies the following conditions: (a) all rows containing only zeroes are at the bottom of the matrix (in the highest-index rows); (b) for every row except zero rows, the leading element is at least one place to the right of the leading element in the previous row. SYMMETRIC MATRIX ──────────────────────────────────────────────────────────── Syntax: m.sym() Returns TRUE if the specified matrix is symmetric. A symmetric matrix is a square matrix that satisfies m(i,j)=m(j,i) for all i and j. ──────────────────────────────────────────────────────────── LIST OF ERROR MESSAGES ──────────────────────────────────────────────────────────── Error messages are printed when a critical error has occured, or when the function called has no other way of informing the program and/or user that an error has occured. Error messages are sent to cerr and can be redirected by the programmer. Not enough memory to allocate vector! Not enough memory to create matrix! ──────────────────────────────────────────────────────────── There is not enough memory for a dynamic allocation during construction of a vector or matrix. If you are using a tiny, small or medium memory model, switch to a larger memory model (registered version only).Try to free up memory by using operators new and delete to construct memory-intensive classes when they are required and destroy them as soon as they are not needed. Avoid large wide-scope objects, including global and static objects. If you are compiling for Windows, increase the size of the local memory heap. If all else fails, you may consider overloading operators new and delete to allocate dynamic memory from EMS or XMS (under DOS), or to allocate memory from the global heap (under Windows). For this to reduce memory use of Dr. Matrix classes, you will need to purchase Doctor Matrix Pro. You will also get this message if you attempt to create a vector or matrix with a non-positive number of rows or columns. Not enough memory! ──────────────────────────────────────────────────────────── This error occurs when an internal dynamic memory allocation has failed, probably due to a shortage of available dynamic memory. See the above error for information on freeing up more memory. Subscript out of range - # Subscript out of range - (#,#) Subscript out of range - Matrix::row(#)! Subscript out of range - Matrix::col(#)! Subscript out of range - Matrix::lead(#)! ──────────────────────────────────────────────────────────── The subscript operator was used to obtain an element of a vector or matrix, but the index was out of range for the specified object. The requested index is indicated. The special forms appear when a function requiring a subscript as a parameter receives an invalid subscript. The requested subscript is indicated, along with the function name and the class of which it is a member. You will also get this error message if you enter a non- positive number as a subscript. Lengths of added vectors must be identical! Lengths of subtracted vectors must be identical! ──────────────────────────────────────────────────────────── Two vectors which were added using the addition operator, or subtracted using the subtraction operator, are not of equal length. Only equal vectors can be added to one another. Orientation of both vectors in vector sum must be identical! Orientation of both vectors in vector subtraction must be identical! ──────────────────────────────────────────────────────────── You attempted to add or subtract two vectors with the same number of items, but with different alignment. Only vectors with the same alignment can be added to or subtracted from one another. To change the alignment of a vector, use the transpose function. Both vectors must be the same length! ──────────────────────────────────────────────────────────── You have attempted to multiply two vectors of different lengths. Only vectors of identical lengths can be multiplied by one another. Orientation of first vector in vector product must be horizontal! Orientation of second vector in vector product must be vertical! ──────────────────────────────────────────────────────────── When multiplying two vectors by one another, the first vector must be aligned horizontally and the second must be aligned vertically. You can change the alignment of a vector using the transpose function. Trace function can only be used on square matrices! ──────────────────────────────────────────────────────────── The trace function tr() can only be performed on square matrices, but you attempted to perform a trace function on a non-square matrix. Added matrices must have identical dimensions! Subtracted matrices must have identical dimensions! ──────────────────────────────────────────────────────────── In order to add or subtract two matrices, they must have the same number of rows and columns. Undefined matrix multiplication! ──────────────────────────────────────────────────────────── You have attempted to multiply two matrices by one another, but the matrices dnot have the correct dimensions in order to be multiplied. The number of columns in the left operand must equal the number of rows in the right operand. Can't calculate determinant of non-square matrix! Attempting to invert a non-square matrix! ──────────────────────────────────────────────────────────── You have called a function which works only on square matrices from a non-square matrix. Attempting to invert a singular matrix! ──────────────────────────────────────────────────────────── You have attempted to invert a singular matrix, but singular matrices are non-invertible. Singular matrices are matrices for which the rank is not equal to the number of rows. You can determine whether a matrix is singular using the sing() function. You can determine whether it is invertible (regular) using the reg() function.