A processor comprising: a decoder to decode a complex matrix conjugation and multiplication instruction including a first source operand to identify a first complex source matrix comprising a first plurality of complex values, a second source operand to identify a second complex source matrix comprising a second plurality of complex values, and a first destination operand to identify a result matrix; execution circuitry (1719, 2091) to execute the complex matrix conjugation and multiplication instruction, the execution circuitry comprising: matrix conjugation hardware logic to determine a plurality of complex conjugate values corresponding to the first plurality of complex values; transpose hardware logic to transpose the plurality of complex conjugate values to generate a conjugate transpose matrix comprising the complex conjugate values; parallel multiplication circuitry to: multiply real values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding imaginary values from the second plurality of complex values to generate a first plurality of imaginary products, and multiply imaginary values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding real values from the second plurality of complex values to generate a second plurality of imaginary products; and addition/subtraction circuitry to add each imaginary product in the first plurality of imaginary products to a corresponding imaginary product in the second plurality of imaginary products to produce a corresponding imaginary component in the result matrix. a decoder to decode a complex matrix conjugation and multiplication instruction including a first source operand to identify a first complex source matrix comprising a first plurality of complex values, a second source operand to identify a second complex source matrix comprising a second plurality of complex values, and a first destination operand to identify a result matrix; execution circuitry (1719, 2091) to execute the complex matrix conjugation and multiplication instruction, the execution circuitry comprising: matrix conjugation hardware logic to determine a plurality of complex conjugate values corresponding to the first plurality of complex values; transpose hardware logic to transpose the plurality of complex conjugate values to generate a conjugate transpose matrix comprising the complex conjugate values; parallel multiplication circuitry to: multiply real values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding imaginary values from the second plurality of complex values to generate a first plurality of imaginary products, and multiply imaginary values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding real values from the second plurality of complex values to generate a second plurality of imaginary products; and addition/subtraction circuitry to add each imaginary product in the first plurality of imaginary products to a corresponding imaginary product in the second plurality of imaginary products to produce a corresponding imaginary component in the result matrix. matrix conjugation hardware logic to determine a plurality of complex conjugate values corresponding to the first plurality of complex values; transpose hardware logic to transpose the plurality of complex conjugate values to generate a conjugate transpose matrix comprising the complex conjugate values; parallel multiplication circuitry to: multiply real values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding imaginary values from the second plurality of complex values to generate a first plurality of imaginary products, and multiply imaginary values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding real values from the second plurality of complex values to generate a second plurality of imaginary products; and multiply real values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding imaginary values from the second plurality of complex values to generate a first plurality of imaginary products, and multiply imaginary values from the plurality of complex conjugate values of the conjugate transpose matrix with corresponding real values from the second plurality of complex values to generate a second plurality of imaginary products; and addition/subtraction circuitry to add each imaginary product in the first plurality of imaginary products to a corresponding imaginary product in the second plurality of imaginary products to produce a corresponding imaginary component in the result matrix.